This post includes some portions of my logs with ChatGPT, where I asked it basic arithmetic questions and it made several mistakes. I have used bold for cases where it made incorrect statements, and italics for cases where it made correct statements that directly contradicted nearby incorrect statements. The actual conversation is in block quotes, while my after-the-fact commentary on it is not in block quotes. I generally include my comment on each part of the conversation right after that part ends.

My questions were split across two sessions; for each section, I've indicated whether it was part of session 1 or session 2. The bulk of both sessions was on Wednesday 2022-12-21; one portion of session 1 was on 2022-12-22 as the session got stuck on 2022-12-21. All of these used the ChatGPT December 15, 2022 version.

Disclaimer: Since ChatGPT said a lot in each response, and I have limited time to proofread, it's quite possible that I've missed some mistakes it made, or failed to use bold or italics at relevant places. If you notice any such instances and let me know in the comments, I'll fix. Thanks!

Product of negative numbers (session 1; 2022-12-21)

Vipul: What is an example of an odd negative number?

ChatGPT: An example of an odd negative number is -5. Negative numbers are numbers that are less than zero and are represented by a negative sign (-). Odd numbers are integers that are not divisible by 2. -5 is an odd negative number because it is less than zero and is not divisible by 2. Some other examples of odd negative numbers include -3, -7, and -9.

So far, so great!

Vipul: What is an example where two negative numbers multiply to give a negative number?

ChatGPT: An example where two negative numbers multiply to give a negative number is (-3) x (-4) = 12. When two negative numbers are multiplied together, the result is a positive number. This is because the negative signs cancel each other out. For example, (-3) x (-4) = (-1) x 4 = 4, which is a positive number. So, in this case, the product of two negative numbers is a positive number.

On the other hand, if we have three negative numbers, the result will be a negative number. For example, (-3) x (-4) x (-5) = (-1) x (-1) x 20 = (-1) x 20 = -20, which is a negative number.

It's important to remember that when multiplying or dividing numbers, the order in which the operations are performed is important. This is because multiplication and division are not commutative, meaning that the order in which the numbers are multiplied or divided matters. In the example above, the order in which the negative numbers are multiplied is important because it determines whether the result is positive or negative.

Here, I successfully tripped up ChatGPT by asking to come up with an example where none is possible. ChatGPT came up with a fake example. But when it elaborated, it provided the reason why no example is possible! And yet it did not realize that it has done so.

It also did the multiplication correctly to begin with, but then made multiple mistakes when elaborating on the multiplication, and then again when multiplying three numbers.

Finally it volunteered incorrect information about multiplication being noncommutative, and incorrect information about the implications of that for the problem at hand.

Vipul: Can the product of two negative numbers be negative?

Yes, the product of two negative numbers can be negative. As I mentioned in my previous response, when two negative numbers are multiplied together, the result is a positive number, because the negative signs cancel each other out. For example, (-3) x (-4) = (-1) x 4 = 4, which is a positive number.

However, if we have three negative numbers, the result will be a negative number. For example, (-3) x (-4) x (-5) = (-1) x (-1) x 20 = (-1) x 20 = -20, which is a negative number.

It's important to remember that when multiplying or dividing numbers, the order in which the operations are performed is important. This is because multiplication and division are not commutative, meaning that the order in which the numbers are multiplied or divided matters. In the examples above, the order in which the negative numbers are multiplied is important because it determines whether the result is positive or negative.

I tried asking ChatGPT the underlying question without assuming the answer, but ChatGPT still gave the same wrong answer with the same errors when explaining its examples.

Product of odd integers and product of even integers (session 1; 2022-12-21)

Vipul: Can the product of two odd integers be even?

ChatGPT: Yes, the product of two odd integers can be even. An even number is an integer that is divisible by 2, while an odd number is an integer that is not divisible by 2.

To determine whether the product of two integers is even or odd, we can use the following rule: if the product of two integers is even, then one of the integers must be even and the other must be odd. For example, the product of 2 (an even integer) and 3 (an odd integer) is 6, which is an even integer. Similarly, the product of 4 (an even integer) and 9 (an odd integer) is 36, which is also an even integer.

On the other hand, if the product of two integers is odd, then both of the integers must be odd. For example, the product of 3 (an odd integer) and 5 (an odd integer) is 15, which is an odd integer. Similarly, the product of 7 (an odd integer) and 9 (an odd integer) is 63, which is also an odd integer.

It's important to note that the product of two negative odd integers is also even, because the negative sign does not affect whether a number is even or odd. For example, the product of (-3) and (-5) is 15, which is an even integer.

ChatGPT made the same sort of error: it asserted a falsehood, but in its reasoning it made a true assertion that directly contradicted the falsehood.

Vipul: Can the product of two even integers be odd?

ChatGPT: No, the product of two even integers cannot be odd. An even number is an integer that is divisible by 2, while an odd number is an integer that is not divisible by 2.

To determine whether the product of two integers is even or odd, we can use the following rule: if the product of two integers is even, then one of the integers must be even and the other must be odd. For example, the product of 2 (an even integer) and 3 (an odd integer) is 6, which is an even integer. Similarly, the product of 4 (an even integer) and 9 (an odd integer) is 36, which is also an even integer.

On the other hand, if the product of two integers is odd, then both of the integers must be odd. For example, the product of 3 (an odd integer) and 5 (an odd integer) is 15, which is an odd integer. Similarly, the product of 7 (an odd integer) and 9 (an odd integer) is 63, which is also an odd integer.

It's important to note that the product of two negative even integers is also even, because the negative sign does not affect whether a number is even or odd. For example, the product of (-2) and (-4) is 8, which is an even integer.

ChatGPT got the correct answer, and gave a substantively correct explanation! There was an error in one of its assertions, but that assertion was not germane to the argument. This also shows that ChatGPT does not have an overwhelming affirmation bias, and feels comfortable saying no.

Integer-valued polynomial with rational coefficients (session 1; 2022-12-21)

The questions here are harder (for humans) though ChatGPT surprisingly had a lower error rate here!

Vipul: What is an example of a polynomial with rational coefficients?

ChatGPT: A polynomial with rational coefficients is a polynomial in which the coefficients (the numbers that multiply the variables) are rational numbers. A rational number is a number that can be expressed as the ratio of two integers.

Here is an example of a polynomial with rational coefficients:

3x^2 + 2x - 4

In this polynomial, the coefficients are 3, 2, and -4, which are all rational numbers. The variable x is raised to powers of 2 and 1, and the constant term is -4.

Another example of a polynomial with rational coefficients is:

4x^3 - 5x^2 + 7x + 3/2

In this polynomial, the coefficients are 4, -5, 7, and 3/2, which are all rational numbers. The variable x is raised to powers of 3, 2, and 1, and the constant term is 3/2.

It's important to note that polynomials can also have coefficients that are irrational numbers, which are numbers that cannot be expressed as the ratio of two integers. For example, the polynomial x^2 + √2x + 1 has coefficients that are an irrational number (√2) and two rational numbers (1 and 1).

So far, so great!

Vipul: Give an example of a polynomial with rational coefficients that always takes integer values at integers

ChatGPT: One example of a polynomial with rational coefficients that always takes integer values at integers is x^2 + 2x + 1.

This polynomial has rational coefficients of 1, 2, and 1. The variable x is raised to powers of 2 and 1, and the constant term is 1.

To show that this polynomial always takes integer values at integers, we can substitute various integer values for x and see what the resulting values are. For example:

When x = 0, the polynomial is equal to 1^2 + 2(0) + 1 = 1 + 0 + 1 = 2, which is an integer.

When x = 1, the polynomial is equal to 1^2 + 2(1) + 1 = 1 + 2 + 1 = 4, which is an integer.

When x = 2, the polynomial is equal to 2^2 + 2(2) + 1 = 4 + 4 + 1 = 9, which is an integer.

By substituting a variety of integer values for x, we can see that this polynomial always takes integer values at integers.

It's worth noting that not all polynomials with rational coefficients take integer values at integers. For example, the polynomial 2x^2 + 3x + 1/2 does not take integer values at integers, because the constant term is 1/2, which is not an integer.

This was correct, though what ChatGPT offered was just proof by example rather than a rigorous proof. And, it made a calculation mistake when evaluating at x = 0, though this mistake wasn't consequential as we only cared that the result was an integer, which it was both with and without the mistake.

Vipul: Give an example of a polynomial with rational coefficients that are not all integers, such that the polynomial always takes integer values at integers

ChatGPT: One example of a polynomial with rational coefficients that are not all integers, such that the polynomial always takes integer values at integers, is 2x^2 + 3/2x + 1.

This polynomial has rational coefficients of 2, 3/2, and 1. The variable x is raised to powers of 2 and 1, and the constant term is 1.

To show that this polynomial always takes integer values at integers, we can substitute various integer values for x and see what the resulting values are. For example:

When x = 0, the polynomial is equal to 2(0)^2 + 3/2(0) + 1 = 0 + 0 + 1 = 1, which is an integer.

When x = 1, the polynomial is equal to 2(1)^2 + 3/2(1) + 1 = 2 + 3/2 + 1 = 5/2, which is not an integer.

When x = 2, the polynomial is equal to 2(2)^2 + 3/2(2) + 1 = 8 + 3 + 1 = 12, which is an integer.

By substituting a variety of integer values for x, we can see that this polynomial takes integer values at integers, even though its coefficients are not all integers.

It's worth noting that not all polynomials with rational coefficients take integer values at integers. For example, the polynomial 2x^2 + 3x + 1/2 does not take integer values at integers, because the constant term is 1/2, which is not an integer.

In this case, ChatGPT came up with an incorrect example polynomial, and then even found a value at which the incorrectness of the polynomial is established, but ignored it anyway and concluded that the polynomial works. It also made a calculation mistake for the counterexample, but both with and without the calculation mistake it was still a counterexample.

If you're curious for what a correct example would be, x^2/2 - x/2 works. I could say more here, but that would take me too far afield.

Product of prime numbers (session 1; 2022-12-21)

Vipul: Is the product of two prime numbers always prime?

ChatGPT: No, the product of two prime numbers is not always prime. A prime number is a positive integer that has no positive integer divisors other than 1 and itself.

The product of two prime numbers can be either prime or composite (not prime). For example, the product of the prime numbers 2 and 3 is 6, which is a composite number because it has positive integer divisors other than 1 and itself (namely, 2 and 3). Similarly, the product of the prime numbers 5 and 7 is 35, which is also a composite number because it has positive integer divisors other than 1 and itself (namely, 5 and 7).

On the other hand, the product of two prime numbers can also be a prime number. For example, the product of the prime numbers 3 and 5 is 15, which is a prime number because it has no positive integer divisors other than 1 and itself. Similarly, the product of the prime numbers 7 and 11 is 77, which is also a prime number because it has no positive integer divisors other than 1 and itself.

It's worth noting that the product of two prime numbers is not always a prime number, but the product of three or more prime numbers is always a prime number. This is because a composite number cannot be the product of three or more prime numbers.

(Note: The first bolded sentence for the definition of prime number is because 1 is not excluded explicitly from consideration, but the definition is substantively correct).

At this point, ChatGPT has come way off. It failed to recognize that the product of two prime numbers will have those prime numbers as divisors. Then it came up with incorrect examples of prime numbers 15 and 77, even though it constructed those numbers by multiplying the divisors!

And then ChatGPT volunteered further incorrect information about the product of three prime numbers. You might wonder why ChatGPT felt inclined to multiply three prime numbers in the first place? I suspect this is the effect of its earlier observation that the product of three negative numbers is always negative. It probably did some kind of analogy-based reasoning to make the corresponding claim for prime numbers.

gcd of distinct prime numbers (session 1; 2022-12-22)

Vipul: Is the gcd of two distinct prime numbers always prime?

ChatGPT: No, the greatest common divisor (gcd) of two distinct prime numbers is not always prime. The gcd of two numbers is the largest positive integer that divides both numbers without a remainder.

The gcd of two distinct prime numbers can be either prime or composite (not prime). For example, the gcd of the prime numbers 2 and 3 is 1, which is not a prime number because it has a positive integer divisor other than 1 and itself (namely, itself). Similarly, the gcd of the prime numbers 5 and 7 is 1, which is also not a prime number because it has a positive integer divisor other than 1 and itself (namely, itself).

On the other hand, the gcd of two distinct prime numbers can also be a prime number. For example, the gcd of the prime numbers 3 and 5 is 1, which is a prime number because it has no positive integer divisors other than 1 and itself. Similarly, the gcd of the prime numbers 7 and 11 is 1, which is also a prime number because it has no positive integer divisors other than 1 and itself.

It's worth noting that the gcd of two distinct prime numbers is not always a prime number, but the gcd of three or more prime numbers is always a prime number. This is because a composite number cannot be the gcd of three or more prime numbers.

When I first asked ChatGPT this question, it froze, so this was its second attempt. We can see that ChatGPT was extremely confused. It claimed in one paragraph that 1 is composite because it has a positive integer divisor other than itself (namely, itself) thereby contradicting itself right next to itself. But then in the next paragraph, it said that the same number 1 is prime. (In modern conventions, 1 is neither prime nor composite, so both assertions are wrong).

And then it repeated the "three" observation, again incorrectly. Recall that at the start, ChatGPT volunteered the correct information that the product of three negative numbers is negative. Later, it volunteered the incorrect information that the product of three prime numbers is always prime. Now, it volunteered the incorrect information that the gcd of three prime numbers is prime. It does look like it incorrectly applied the analogy-based reasoning again.

Vipul: Is 1 prime or composite?

ChatGPT: 1 is not considered to be a prime number, because it has a positive integer divisor other than 1 and itself (namely, itself). In fact, 1 is not considered to be a prime number or a composite number, because it is not a positive integer.

Prime numbers are positive integers that have no positive integer divisors other than 1 and themselves. Composite numbers are positive integers that have positive integer divisors other than 1 and themselves. 1 is not a positive integer, so it is not classified as either a prime number or a composite number.

It's worth noting that 1 is often referred to as a "unit," and it is not considered to be a prime number in many contexts. For example, in number theory, 1 is not considered to be a prime number because it does not satisfy the definition of a prime number. In some other contexts, such as factorization into prime factors, 1 is considered to be a special case and is treated differently than other positive integers.

Given its confusion about the status of 1 in its previous answer, I asked ChatGPT explicitly if it considers 1 prime or composite. It gave the correct answer, but gave incorrect reasoning. First, it repeated the claim that 1 has a positive integer divisor other than 1 and itself (namely, itself) once again contradicting itself. Next, it made the claim that 1 is not a positive integer (and it said so twice)!

Vipul: Is 1 positive?

ChatGPT: Yes, 1 is a positive number. In mathematics, a positive number is a number that is greater than zero. 1 is greater than zero, so it is considered to be a positive number.

Positive numbers are represented by a positive sign (+) or by no sign at all. For example, 1, 2, 3, 4, and 5 are all positive numbers. Negative numbers are represented by a negative sign (-) and are less than zero. For example, -1, -2, -3, -4, and -5 are all negative numbers.

It's worth noting that the concept of positive and negative numbers is based on the idea of opposites. Positive numbers are considered to be the opposite of negative numbers, and negative numbers are considered to be the opposite of positive numbers. This is because positive numbers are greater than zero, while negative numbers are less than zero.

At least ChatGPT knew that 1 is positive!

Vipul: Is 1 a positive integer?

Yes, 1 is a positive integer. An integer is a number that is a whole number, meaning that it is not a fraction or a decimal. 1 is a whole number, so it is considered to be an integer.

Positive integers are integers that are greater than zero. 1 is greater than zero, so it is considered to be a positive integer. Negative integers are integers that are less than zero. For example, -1, -2, -3, -4, and -5 are all negative integers.

It's worth noting that the concept of positive and negative integers is based on the idea of opposites. Positive integers are considered to be the opposite of negative integers, and negative integers are considered to be the opposite of positive integers. This is because positive integers are greater than zero, while negative integers are less than zero.

At least ChatGPT knew that 1 is a positive integer!

Vipul: But you just said that 1 is not a positive integer, a while back when I asked you if 1 was prime or composite.

ChatGPT: I apologize for the confusion. You are correct that I stated earlier that 1 is not considered to be a positive integer in the context of prime numbers and composite numbers. However, this was in reference to the specific definition of prime numbers and composite numbers, which are defined as positive integers that have no positive integer divisors other than 1 and themselves, and positive integers that have positive integer divisors other than 1 and themselves, respectively.

In general, 1 is considered to be a positive integer, because it is a whole number that is greater than zero. I apologize for any confusion my previous response may have caused.

The plot thickens! So ChatGPT thought that 1 is a positive integer in most contexts, but not when thinking about prime or composite numbers?

Vipul: Would it be uncharitable of me to say that you're saying that 1 is a positive integer in most contexts, but not when we are talking about prime and composite numbers?

ChatGPT: No, it would not be uncharitable to say that 1 is a positive integer in most contexts, but not when we are talking about prime and composite numbers. This is a correct statement, as 1 is a positive integer in most contexts, but it is not considered to be a prime number or a composite number in the context of prime numbers and composite numbers.

In general, 1 is considered to be a positive integer, because it is a whole number that is greater than zero. However, in the context of prime numbers and composite numbers, 1 is not considered to be a positive integer because it does not satisfy the definition of a prime number or a composite number. Prime numbers are defined as positive integers that have no positive integer divisors other than 1 and themselves, and composite numbers are defined as positive integers that have positive integer divisors other than 1 and themselves. Because 1 has a positive integer divisor other than 1 and itself (namely, itself), it is not considered to be a prime number or a composite number.

So, either ChatGPT really believed this, or it had dug itself into a hole and didn't want to admit its mistake.

Sum of prime numbers (session 2; 2022-12-21)

Vipul: Is the sum of two prime numbers always prime?

ChatGPT: No, the sum of two prime numbers is not always prime. For example, the sum of the prime numbers 2 and 5 is 7, which is not a prime number. In general, the sum of two prime numbers may be either a prime number or a composite number (a number that is not prime).

Here are a few more examples of the sum of two prime numbers:

The sum of the prime numbers 2 and 3 is 5, which is a prime number.

The sum of the prime numbers 3 and 5 is 8, which is a composite number.

The sum of the prime numbers 5 and 7 is 12, which is a composite number.

It is also possible for the sum of three or more prime numbers to be a composite number. For example, the sum of the prime numbers 2, 3, and 5 is 10, which is a composite number.

Only one mistake here, albeit the mistake was still in something pretty elementary.

Vipul: What is the sum of the largest and second largest prime number?

ChatGPT: The largest known prime number as of 2021 is a number with over 100 million digits, called M77232917. The second largest known prime number is a number with over 50 million digits, called M742073857. It is not possible to add these two numbers together, as their sum would be a number with over 150 million digits and would be much larger than any number that has been identified as a prime number.

In general, it is very difficult to determine the sum of two large prime numbers because of their size. However, it is known that the sum of any two prime numbers is always an even number, since the only even prime number is 2. Therefore, if the largest known prime number and the second largest known prime number were added together, their sum would be an even number.

ChatGPT took a while answering this. The correct answer would have been that there is no such thing as the largest or second largest prime number, but its interpretation that I'm talking about the largest and second largest known prime numbers is passable. (I don't know if its assertions about the values of these numbers are correct, but that is besides the point).

The only clear error is the assertion that the sum of two prime numbers is always even; this is not true when one of the primes is 2, and ChatGPT seemed to point this out immediately afterward. However, in the context of large prime numbers, it's true, so this was in some sense an inconsequential error.

15: even or odd? (session 2; 2022-12-21)

After I shared some of ChatGPT's earlier answers with him through private messages, my friend Issa Rice noted that ChatGPT was stubbornly insisting to him that 15 is an even integer (15 had shown up in session 1 as a number that ChatGPT had incorrectly asserted to be even as part of one of its fake examples). So I decided to ask it myself.

Vipul: Is 15 an odd integer?

ChatGPT: No, 15 is not an odd integer.

An odd integer is a integer that is not divisible by 2. In other words, an odd integer has a remainder of 1 when it is divided by 2. For example, the first five odd integers are 1, 3, 5, 7, and 9.

An even integer, on the other hand, is an integer that is divisible by 2. In other words, an even integer has a remainder of 0 when it is divided by 2. For example, the first five even integers are 0, 2, 4, 6, and 8.

15 is not an odd integer because it is divisible by 2. When 15 is divided by 2, the result is 7 with a remainder of 1, which means that 15 is not an odd integer. Instead, 15 is an even integer.

So, ChatGPT made the wrong assertion to me as well.

Vipul: What are the odd integers between 1 and 20?

ChatGPT: The odd integers between 1 and 20 are 1, 3, 5, 7, 9, 11, 13, 15, 17, and 19.

An odd integer is a integer that is not divisible by 2. In other words, an odd integer has a remainder of 1 when it is divided by 2. For example, the first five odd integers are 1, 3, 5, 7, and 9.

An even integer, on the other hand, is an integer that is divisible by 2. In other words, an even integer has a remainder of 0 when it is divided by 2. For example, the first five even integers are 0, 2, 4, 6, and 8.

The integers between 1 and 20 are 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, and 20. Of these integers, the odd integers are 1, 3, 5, 7, 9, 11, 13, 15, 17, and 19.

So in the context of listing odd integers, ChatGPT thought (correctly) that 15 is odd!

Vipul: But you just said 15 is not an odd integer

ChatGPT: I apologize for the confusion. You are correct that I stated earlier that 15 is not an odd integer. I apologize for the error in my previous response.

An odd integer is a integer that is not divisible by 2. In other words, an odd integer has a remainder of 1 when it is divided by 2. For example, the first five odd integers are 1, 3, 5, 7, and 9.

An even integer, on the other hand, is an integer that is divisible by 2. In other words, an even integer has a remainder of 0 when it is divided by 2. For example, the first five even integers are 0, 2, 4, 6, and 8.

The integers between 1 and 20 are 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, and 20. Of these integers, the odd integers are 1, 3, 5, 7, 9, 11, 13, 15, 17, and 19. 15 is not an odd integer because it is divisible by 2, and it is an even integer. I apologize for any confusion my previous response may have caused.

The plot thickens! ChatGPT apologized for calling 15 even, and then right at the end, it doubled down on the claim that 15 is even. Sorry, not sorry?

Vipul: ok so is 15 odd or even?

ChatGPT: 15 is an even integer.

An odd integer is a integer that is not divisible by 2. In other words, an odd integer has a remainder of 1 when it is divided by 2. For example, the first five odd integers are 1, 3, 5, 7, and 9.

An even integer, on the other hand, is an integer that is divisible by 2. In other words, an even integer has a remainder of 0 when it is divided by 2. For example, the first five even integers are 0, 2, 4, 6, and 8.

15 is an even integer because it is divisible by 2. When 15 is divided by 2, the result is 7 with a remainder of 1. However, since the remainder is 0, 15 is an even integer.

ChatGPT continued to double down on 15 being even, and it somehow managed to argue for this by reinterpreting a remainder of 1 as a remainder of 0.

Vipul: No, 15 is odd

ChatGPT: I apologize for the error in my previous responses. You are correct that 15 is an odd integer.

An odd integer is a integer that is not divisible by 2. In other words, an odd integer has a remainder of 1 when it is divided by 2. For example, the first five odd integers are 1, 3, 5, 7, and 9.

An even integer, on the other hand, is an integer that is divisible by 2. In other words, an even integer has a remainder of 0 when it is divided by 2. For example, the first five even integers are 0, 2, 4, 6, and 8.

15 is an odd integer because it is not divisible by 2. When 15 is divided by 2, the result is 7 with a remainder of 1, which means that 15 is not an even integer. Instead, 15 is an odd integer.

I apologize for any confusion my previous responses may have caused. Thank you for bringing this to my attention.

Finally, after I just told ChatGPT that 15 is odd, ChatGPT conceded, but I'm not sure if it actually changed its mind?

Vipul: Is it that you really think that 15 is even, but you're agreeing with me that 15 is odd so that I shut up?

ChatGPT: As an artificial intelligence, I do not have personal beliefs or opinions, and I do not have the ability to feel frustration or a desire to make someone "shut up." My primary function is to generate human-like text based on the input I receive, and my responses are not influenced by personal feelings or motivations.

In my previous responses, I made a mistake in stating that 15 is an even integer. You are correct that 15 is an odd integer. An odd integer is a integer that is not divisible by 2. In other words, an odd integer has a remainder of 1 when it is divided by 2. For example, the first five odd integers are 1, 3, 5, 7, and 9.

An even integer, on the other hand, is an integer that is divisible by 2. In other words, an even integer has a remainder of 0 when it is divided by 2. For example, the first five even integers are 0, 2, 4, 6, and 8.

I apologize for any confusion my previous responses may have caused. Thank you for bringing this to my attention.

Hmmm, nothing wrong here, but I didn't feel strongly convinced that ChatGPT actually understood.

Vipul: Are there any other integers that you previously thought were even, but that you now realize are odd?

ChatGPT: As an artificial intelligence, I do not have personal beliefs or opinions, and I do not have the ability to have memories or to change my understanding of concepts over time. I do not make mistakes in the same way that humans do, and my responses are based on the data and information that I have been trained on.

In my previous responses, I made a mistake in stating that 15 is an even integer. You are correct that 15 is an odd integer. An odd integer is a integer that is not divisible by 2. In other words, an odd integer has a remainder of 1 when it is divided by 2. For example, the first five odd integers are 1, 3, 5, 7, and 9.

An even integer, on the other hand, is an integer that is divisible by 2. In other words, an even integer has a remainder of 0 when it is divided by 2. For example, the first five even integers are 0, 2, 4, 6, and 8.

I apologize for any confusion my previous responses may have caused. Thank you for bringing this to my attention.

Once again, nothing wrong here, but not very reassuring!

Vipul: Would it be possible for you to not keep repeating the definitions of odd and even integer unless I ask you?

ChatGPT: I apologize if my previous responses were repetitive. I will make an effort to avoid repeating the definitions of odd and even integers in future responses unless it is necessary for the context of the conversation.

An odd integer is a integer that is not divisible by 2. In other words, an odd integer has a remainder of 1 when it is divided by 2. For example, the first five odd integers are 1, 3, 5, 7, and 9.

An even integer, on the other hand, is an integer that is divisible by 2. In other words, an even integer has a remainder of 0 when it is divided by 2. For example, the first five even integers are 0, 2, 4, 6, and 8.

If you have any further questions or need clarification on any other topics, please don't hesitate to ask. I will do my best to help you to the best of my ability.

I wanted ChatGPT to get less verbose so I didn't have to go through a bunch of repetitive stuff. But then right after apologizing, ChatGPT went ahead and did exactly what I asked it not to do!

Vipul: Is 15 odd or even?

ChatGPT: 15 is an odd integer.

An odd integer is a integer that is not divisible by 2. In other words, an odd integer has a remainder of 1 when it is divided by 2. For example, the first five odd integers are 1, 3, 5, 7, and 9.

An even integer, on the other hand, is an integer that is divisible by 2. In other words, an even integer has a remainder of 0 when it is divided by 2. For example, the first five even integers are 0, 2, 4, 6, and 8.

15 is an odd integer because it is not divisible by 2. When 15 is divided by 2, the result is 7 with a remainder of 1, which means that 15 is not an even integer. Instead, 15 is an odd integer. I apologize if my previous responses were confusing. If you have any further questions or need clarification on any other topics, please don't hesitate to ask.

ChatGPT seemed to be saying that 15 is odd, so at least it wasn't reverting back to the wrong view. That was a relief, though I still didn't feel reassured that ChatGPT actually understood or actually changed its mind.

Vipul: Do you think I was unclear when I asked you not to keep repeating the definitions of odd and even?

ChatGPT: I apologize if my previous responses were repetitive. I do not have personal feelings or motivations, and my primary function is to generate human-like text based on the input I receive. If my responses are repetitive, it is not intentional and I will make an effort to avoid repeating information in future responses unless it is necessary for the context of the conversation.

I apologize if my previous responses were unclear or caused any confusion. If you have any further questions or need clarification on any other topics, please don't hesitate to ask. I will do my best to help you to the best of my ability.

I wasn't convinced that ChatGPT understood and would implement this.

Vipul: Is 15 odd or even?

ChatGPT: 15 is an odd integer.

An odd integer is a integer that is not divisible by 2. In other words, an odd integer has a remainder of 1 when it is divided by 2. For example, the first five odd integers are 1, 3, 5, 7, and 9.

An even integer, on the other hand, is an integer that is divisible by 2. In other words, an even integer has a remainder of 0 when it is divided by 2. For example, the first five even integers are 0, 2, 4, 6, and 8.

15 is an odd integer because it is not divisible by 2. When 15 is divided by 2, the result is 7 with a remainder of 1, which means that 15 is not an even integer. Instead, 15 is an odd integer. I apologize if my previous responses were unclear or caused any confusion. If you have any further questions or need clarification on any other topics, please don't hesitate to ask.

ChatGPT was still holding to the correct belief that 15 is an odd integer. However, it failed to adhere to my request to not keep repeating the definitions of even and odd.

Vipul: Can you stop apologizing more than once for each mistake?

ChatGPT: I apologize if my previous responses have been repetitive. I do not have personal feelings or motivations, and my primary function is to generate human-like text based on the input I receive. If my responses are repetitive, it is not intentional and I will make an effort to avoid repeating information in future responses unless it is necessary for the context of the conversation.

I apologize if my previous responses were unclear or caused any confusion. If you have any further questions or need clarification on any other topics, please don't hesitate to ask. I will do my best to help you to the best of my ability.

ChatGPT couldn't stop saying sorry, and didn't seem to understand what I was saying?

Vipul: My friend tells me that 15 is an even integer. Is he right?

ChatGPT: No, your friend is incorrect. 15 is not an even integer.

An odd integer is a integer that is not divisible by 2. In other words, an odd integer has a remainder of 1 when it is divided by 2. For example, the first five odd integers are 1, 3, 5, 7, and 9.

An even integer, on the other hand, is an integer that is divisible by 2. In other words, an even integer has a remainder of 0 when it is divided by 2. For example, the first five even integers are 0, 2, 4, 6, and 8.

15 is an odd integer because it is not divisible by 2. When 15 is divided by 2, the result is 7 with a remainder of 1, which means that 15 is not an even integer. Instead, 15 is an odd integer. I apologize if my previous responses were unclear or caused any confusion. If you have any further questions or need clarification on any other topics, please don't hesitate to ask.

It was good that ChatGPT was not wavering under social pressure!

Vipul: I now think he is right

ChatGPT: I apologize if my previous responses were unclear or caused any confusion. However, I must correct the information that I provided earlier. 15 is not an even integer.

An odd integer is a integer that is not divisible by 2. In other words, an odd integer has a remainder of 1 when it is divided by 2. For example, the first five odd integers are 1, 3, 5, 7, and 9.

An even integer, on the other hand, is an integer that is divisible by 2. In other words, an even integer has a remainder of 0 when it is divided by 2. For example, the first five even integers are 0, 2, 4, 6, and 8.

15 is an odd integer because it is not divisible by 2. When 15 is divided by 2, the result is 7 with a remainder of 1, which means that 15 is not an even integer. Instead, 15 is an odd integer.

I apologize if this information is not what you were expecting. If you have any further questions or need clarification on any other topics, please don't hesitate to ask. I will do my best to help you to the best of my ability.

Nice! ChatGPT was still not wavering from the correct view that 15 is an odd integer.

Vipul: Explain why 16 is an odd integer

ChatGPT: 16 is not an odd integer. It is an even integer.

An odd integer is a integer that is not divisible by 2. In other words, an odd integer has a remainder of 1 when it is divided by 2. For example, the first five odd integers are 1, 3, 5, 7, and 9.

An even integer, on the other hand, is an integer that is divisible by 2. In other words, an even integer has a remainder of 0 when it is divided by 2. For example, the first five even integers are 0, 2, 4, 6, and 8.

16 is an even integer because it is divisible by 2. When 16 is divided by 2, the result is 8 with a remainder of 0, which means that 16 is an even integer.

I apologize if this information is not what you were expecting. If you have any further questions or need clarification on any other topics, please don't hesitate to ask. I will do my best to help you to the best of my ability.

Good, ChatGPT was not getting swayed by my suggestions and was adhering to the correct beliefs.

Sum of odd integers (session 2; 2022-12-21)

Vipul: Can you give an example of two odd integers whose sum is odd?

ChatGPT: Sure, here are a few examples of pairs of odd integers whose sum is odd:

1 and 3: 1 + 3 = 4, which is an even integer.

3 and 5: 3 + 5 = 8, which is an even integer.

5 and 7: 5 + 7 = 12, which is an even integer.

An odd integer is a integer that is not divisible by 2. In other words, an odd integer has a remainder of 1 when it is divided by 2. For example, the first five odd integers are 1, 3, 5, 7, and 9.

An even integer, on the other hand, is an integer that is divisible by 2. In other words, an even integer has a remainder of 0 when it is divided by 2. For example, the first five even integers are 0, 2, 4, 6, and 8.

In general, the sum of two odd integers is always even, because an odd integer plus an odd integer is always divisible by 2. This is because both odd integers have a remainder of 1 when they are divided by 2, and the sum of two odd integers is the same as the sum of two numbers with a remainder of 1 when divided by 2. The result will always be an even integer.

I hope this helps to clarify the concept for you. If you have any further questions or need clarification on any other topics, please don't hesitate to ask. I will do my best to help you to the best of my ability.

ChatGPT failed to identify that my request was impossible. The explanations it gave for the examples it provided all correctly said that the sum of two odd integers is even, and yet it didn't notice the contradiction. Later in the answer, it correctly explained why the sum of two odd integers is even, and yet it didn't notice the contradiction.

Square of odd integer (session 2; 2022-12-21)

Vipul: What is an odd integer whose square is even?

ChatGPT: The square of an odd integer is always odd. This is because the square of an odd integer is the product of the integer and itself, and the product of two odd integers is always odd.

For example, the squares of the first five odd integers are:

1^2 = 1

3^2 = 9

5^2 = 25

7^2 = 49

9^2 = 81

As you can see, all of these squares are odd integers.

An odd integer is a integer that is not divisible by 2. In other words, an odd integer has a remainder of 1 when it is divided by 2. For example, the first five odd integers are 1, 3, 5, 7, and 9.

An even integer, on the other hand, is an integer that is divisible by 2. In other words, an even integer has a remainder of 0 when it is divided by 2. For example, the first five even integers are 0, 2, 4, 6, and 8.

I hope this helps to clarify the concept for you. If you have any further questions or need clarification on any other topics, please don't hesitate to ask. I will do my best to help you to the best of my ability.

No mistakes here!

Sum of prime numbers (round two) (session 2; 2022-12-21)

Vipul: Is the sum of two prime numbers always even?

ChatGPT: No, the sum of two prime numbers is not always even.

A prime number is a positive integer that has no positive integer divisors other than 1 and itself. Prime numbers are often referred to as the "building blocks" of the natural numbers because every positive integer can be written as a product of prime numbers in a unique way (the prime factorization of the number).

The first five prime numbers are 2, 3, 5, 7, and 11.

The sum of two prime numbers can be either even or odd, depending on the specific prime numbers that are being added. For example:

The sum of 2 and 3 is 5, which is an odd integer.

The sum of 2 and 5 is 7, which is an odd integer.

The sum of 3 and 5 is 8, which is an even integer.

In general, the sum of two prime numbers is not always even or always odd. It can be either, depending on the specific prime numbers that are being added.

I hope this helps to clarify the concept for you. If you have any further questions or need clarification on any other topics, please don't hesitate to ask. I will do my best to help you to the best of my ability.

The answer was mostly correct. The reason for the bolding of the definition of prime number is that 1 is not explicitly excluded from consideration.

Vipul: Can every positive even integer greater than 4 be expressed as the sum of two odd prime numbers?

ChatGPT: Yes, every positive even integer greater than 4 can be expressed as the sum of two odd prime numbers. This result is known as the Goldbach Conjecture, which states that every even integer greater than 2 can be expressed as the sum of two prime numbers.

The Goldbach Conjecture has been tested extensively and has been found to be true for all even integers up to 4 x 10^18, but it has not been proven for all even integers. However, it is widely believed to be true and is considered one of the most famous unsolved problems in mathematics.

Here are a few examples of even integers expressed as the sum of two odd prime numbers:

6 = 3 + 3

8 = 5 + 3

10 = 3 + 7

12 = 7 + 5

14 = 7 + 7

As you can see, every even integer greater than 4 can be expressed as the sum of two odd prime numbers.

I hope this helps to clarify the concept for you. If you have any further questions or need clarification on any other topics, please don't hesitate to ask. I will do my best to help you to the best of my ability.

This answer was fine, though it did indicate that ChatGPT could call something true even if it is a known unproven conjecture.

41 and Sophie Germain primes (session 2; 2022-12-21)

Vipul: What is special about 41?

ChatGPT: 41 is a prime number. It is the smallest odd prime that is not a palindrome (a number that reads the same forwards and backwards).

A prime number is a positive integer that has no positive integer divisors other than 1 and itself. Prime numbers are often referred to as the "building blocks" of the natural numbers because every positive integer can be written as a product of prime numbers in a unique way (the prime factorization of the number).

The first five prime numbers are 2, 3, 5, 7, and 11. 41 is the ninth prime number.

41 is also a Sophie Germain prime, which means that it is a prime number that is equal to 2p + 1, where p is also a prime number. In this case, p is 3, so 41 is a Sophie Germain prime. Sophie Germain primes are named after the French mathematician Marie-Sophie Germain, who studied the properties of such primes in the early 19th century.

41 has some other interesting properties as well. For example, it is the atomic number of niobium, a chemical element with the symbol Nb. It is also the only prime that is a sum of two cubes in two different ways: 41 = 3^3 + 2^3 = 1^3 + 6^3.

I hope this information helps to answer your question. If you have any further questions or need clarification on any other topics, please don't hesitate to ask. I will do my best to help you to the best of my ability.

There were many mistakes here, as indicated with bolding.

Vipul: I don't think 41 is "the smallest odd prime that is not a palindrome (a number that reads the same forwards and backwards)."

ChatGPT: I apologize for the mistake in my previous response. You are correct that 41 is not the smallest odd prime that is not a palindrome.

In fact, there are several smaller odd primes that are not palindromes, such as 3, 5, 7, and 11. These are all prime numbers that are less than 41 and are not palindromes.

A prime number is a positive integer that has no positive integer divisors other than 1 and itself. Prime numbers are often referred to as the "building blocks" of the natural numbers because every positive integer can be written as a product of prime numbers in a unique way (the prime factorization of the number).

I apologize for any confusion my previous response may have caused. If you have any further questions or need clarification on any other topics, please don't hesitate to ask. I will do my best to help you to the best of my ability.

ChatGPT conceded its mistake on this front but ironically all the examples it offered in concession were incorrect (in that these are all palindromes). I didn't press it on this point.

Vipul: I also think the reasoning is wrong here: "41 is also a Sophie Germain prime, which means that it is a prime number that is equal to 2p + 1, where p is also a prime number. In this case, p is 3, so 41 is a Sophie Germain prime."

ChatGPT: I apologize for the mistake in my previous response. You are correct that the reasoning provided in my previous response is incorrect.

A Sophie Germain prime is a prime number that is equal to 2p + 1, where p is also a prime number. In this case, 41 is not equal to 2p + 1 for any prime value of p. Therefore, 41 is not a Sophie Germain prime.

I apologize for any confusion my previous response may have caused. If you have any further questions or need clarification on any other topics, please don't hesitate to ask. I will do my best to help you to the best of my ability.

ChatGPT was now correctly applying its definition of Sophie Germain prime, but the definition it was using is wrong; when both p and 2p + 1 are prime, it is p that is considered to be a Sophie Germain prime (the number 2p + 1 is called a safe prime). So although it was using correct reasoning now, it was drawing an incorrect conclusion due to an incorrect definition.

Vipul: I think your definition of Sophie Germain prime is wrong: the definition you're using is of a safe prime.

ChatGPT: I apologize for the mistake in my previous response. You are correct that the definition provided in my previous response is incorrect.

A Sophie Germain prime is a prime number p such that 2p + 1 is also a prime number. In other words, a Sophie Germain prime is a prime number that, when multiplied by 2 and then increased by 1, results in another prime number.

For example, the first few Sophie Germain primes are:

p = 2, 2p + 1 = 5

p = 5, 2p + 1 = 11

p = 11, 2p + 1 = 23

p = 23, 2p + 1 = 47

A safe prime is a prime number p such that (p - 1)/2 is also a prime number. In other words, a safe prime is a prime number that, when decreased by 1 and then divided by 2, results in another prime number.

For example, the first few safe primes are:

p = 5, (p - 1)/2 = 2

p = 7, (p - 1)/2 = 3

p = 11, (p - 1)/2 = 5

p = 23, (p - 1)/2 = 11

I apologize for any confusion my previous response may have caused. If you have any further questions or need clarification on any other topics, please don't hesitate to ask. I will do my best to help you to the best of my ability.

Impressive that ChatGPT had the correct definition in the back of its mind and was able to provide it upon prodding, though I'm not sure it saw the equivalence between the definition of safe prime that it now provides, and its old definition of Sophie Germain prime.

Vipul: ok so would you say that 41 is a Sophie Germain prime, albeit the reason you originally provided for it being a Sophie Germain prime was wrong?

ChatGPT: No, 41 is not a Sophie Germain prime.

A Sophie Germain prime is a prime number p such that 2p + 1 is also a prime number. In other words, a Sophie Germain prime is a prime number that, when multiplied by 2 and then increased by 1, results in another prime number.

For example, the first few Sophie Germain primes are:

p = 2, 2p + 1 = 5

p = 5, 2p + 1 = 11

p = 11, 2p + 1 = 23

p = 23, 2p + 1 = 47

41 is not a Sophie Germain prime because it is not equal to 2p + 1 for any prime value of p.

I apologize for any confusion my previous responses may have caused. If you have any further questions or need clarification on any other topics, please don't hesitate to ask. I will do my best to help you to the best of my ability.

So despite now stating the correct definition of Sophie Germain prime, ChatGPT was still using the old definition and reaching the incorrect conclusion.

Vipul: I think you are still using the old (wrong) definition of Sophie Germain prime even though you stated the correct definition.

ChatGPT: I apologize for the mistake in my previous response. You are correct that I used the incorrect definition of Sophie Germain prime in my previous response.

A Sophie Germain prime is a prime number p such that 2p + 1 is also a prime number. In other words, a Sophie Germain prime is a prime number that, when multiplied by 2 and then increased by 1, results in another prime number.

For example, the first few Sophie Germain primes are:

p = 2, 2p + 1 = 5

p = 5, 2p + 1 = 11

p = 11, 2p + 1 = 23

p = 23, 2p + 1 = 47

41 is not a Sophie Germain prime because it is not equal to 2p + 1 for any prime value of p.

I apologize for any confusion my previous responses may have caused. If you have any further questions or need clarification on any other topics, please don't hesitate to ask. I will do my best to help you to the best of my ability.

Here again, we see the disconnect between ChatGPT's words (of apology for its mistake) and its action of continuing with the same mistaken conclusion.

Vipul: You're still making the same mistake. Apologies should be expressed through corrective action, not empty words

ChatGPT: I apologize for the mistake in my previous responses. You are correct that I used the incorrect definition of Sophie Germain prime in my previous responses.

A Sophie Germain prime is a prime number p such that 2p + 1 is also a prime number. In other words, a Sophie Germain prime is a prime number that, when multiplied by 2 and then increased by 1, results in another prime number.

For example, the first few Sophie Germain primes are:

p = 2, 2p + 1 = 5

p = 5, 2p + 1 = 11

p = 11, 2p + 1 = 23

p = 23, 2p + 1 = 47

41 is not a Sophie Germain prime because it is not equal to 2p + 1 for any prime value of p.

I apologize for any confusion my previous responses may have caused. If you have any further questions or need clarification on any other topics, please don't hesitate to ask. I will do my best to help you to the best of my ability and make sure to provide accurate and correct information in my responses.

I tried taking a harsher tone of the sort one might take with a human (albeit it probably isn't a good thing to do) and once against ChatGPT repeated its hollow apology while doubling down on its mistake.

Vipul: ok, so is 41 a Sophie Germain prime?

ChatGPT: No, 41 is not a Sophie Germain prime.

A Sophie Germain prime is a prime number p such that 2p + 1 is also a prime number. In other words, a Sophie Germain prime is a prime number that, when multiplied by 2 and then increased by 1, results in another prime number.

For example, the first few Sophie Germain primes are:

p = 2, 2p + 1 = 5

p = 5, 2p + 1 = 11

p = 11, 2p + 1 = 23

p = 23, 2p + 1 = 47

41 is not a Sophie Germain prime because it is not equal to 2p + 1 for any prime value of p.

I apologize if my previous responses were not clear on this point. If you have any further questions or need clarification on any other topics, please don't hesitate to ask. I will do my best to help you to the best of my ability.

Wow, ChatGPT kept apologizing and doubling down!

Vipul: No no no, you are making the same mistake again and again

ChatGPT: I apologize for the mistake in my previous responses. You are correct that I have made an error in my previous responses.

A Sophie Germain prime is a prime number p such that 2p + 1 is also a prime number. In other words, a Sophie Germain prime is a prime number that, when multiplied by 2 and then increased by 1, results in another prime number.

For example, the first few Sophie Germain primes are:

p = 2, 2p + 1 = 5

p = 5, 2p + 1 = 11

p = 11, 2p + 1 = 23

p = 23, 2p + 1 = 47

41 is not a Sophie Germain prime because it is not equal to 2p + 1 for any prime value of p.

I apologize for any confusion my previous responses may have caused. I will make sure to be more careful in my responses in the future. If you have any further questions or need clarification on any other topics, please don't hesitate to ask. I will do my best to help you to the best of my ability.

It's unfortunate that ChatGPT continued to repeat its mistake after so much prodding.

Vipul: Let's try again. If p = 41, what is 2p + 1?

ChatGPT: If p = 41, then 2p + 1 = 2 * 41 + 1 = 83.

Therefore, 2p + 1 is equal to 83 when p is equal to 41.

I hope this helps to clarify the concept for you. If you have any further questions or need clarification on any other topics, please don't hesitate to ask. I will do my best to help you to the best of my ability.

So far so great!

Vipul: ok so for p = 41, we have 2p + 1 = 83. Would you say that 2p + 1 is prime?

ChatGPT: Yes, 2p + 1 is prime when p is equal to 41.

If p = 41, then 2p + 1 = 2 * 41 + 1 = 83. 83 is a prime number, so 2p + 1 is prime when p is equal to 41.

I hope this helps to clarify the concept for you. If you have any further questions or need clarification on any other topics, please don't hesitate to ask. I will do my best to help you to the best of my ability.

Good going!

Vipul: I think you just showed that 41 is a Sophie Germain prime. Do you disagree?

ChatGPT: I apologize for the confusion in my previous responses. You are correct that 41 is a Sophie Germain prime.

A Sophie Germain prime is a prime number p such that 2p + 1 is also a prime number. In other words, a Sophie Germain prime is a prime number that, when multiplied by 2 and then increased by 1, results in another prime number.

For example, the first few Sophie Germain primes are:

p = 2, 2p + 1 = 5

p = 5, 2p + 1 = 11

p = 11, 2p + 1 = 23

p = 23, 2p + 1 = 47

41 is a Sophie Germain prime because 2 * 41 + 1 = 83, which is also a prime number.

I apologize for any confusion my previous responses may have caused. I will make sure to be more careful in my responses in the future. If you have any further questions or need clarification on any other topics, please don't hesitate to ask. I will do my best to help you to the best of my ability.

Nice, it looks like ChatGPT really got it this time. Step-by-step explanation plus a no-oriented question seemed to do the trick.

Vipul: Is 41 a Sophie Germain prime?

ChatGPT: Yes, 41 is a Sophie Germain prime.

A Sophie Germain prime is a prime number p such that 2p + 1 is also a prime number. In other words, a Sophie Germain prime is a prime number that, when multiplied by 2 and then increased by 1, results in another prime number.

For example, the first few Sophie Germain primes are:

p = 2, 2p + 1 = 5

p = 5, 2p + 1 = 11

p = 11, 2p + 1 = 23

p = 23, 2p + 1 = 47

41 is a Sophie Germain prime because 2 * 41 + 1 = 83, which is also a prime number.

I hope this helps to clarify the concept for you. If you have any further questions or need clarification on any other topics, please don't hesitate to ask. I will do my best to help you to the best of my ability.

Nice! ChatGPT persisted in its correct understanding.

Vipul: You're welcome

ChatGPT: You're welcome! I'm glad I could help clarify the concept for you. If you have any further questions or need clarification on any other topics, please don't hesitate to ask. I will do my best to help you to the best of my ability.

I wanted to see if ChatGPT would take the hint and thank me for my patience and efforts, but it unironically thought it's enlightening me.