Whilst watching the footie, I had been thinking about alternative bases for arithmetic, and started a ChatGPT chat about it. If you ignore the false start at the top, then the definition of "special division" that I created seems to have piqued ChatGPT's interest. The conversation went much further than I expected. I'm posting the transcript to invite comments on whether this is interesting or merely ChatGPT going off on a tangent.

ChatGPT cottoned on very well to my actual intention, which was to invent a different starting point for doing certain math things, whose concepts would make it easier to solve certain problems in an elegant way. The definition of special division is arbitrary but turned out to have certain properties that make it interesting (as discussed in the initial phases of the full chat.)

Full thread: https://chatgpt.com/share/6abc3dbf-e790-83eb-8b5b-2f478a925703

ChatGPT's précis (skip to point 20 for a tl;dr):

Starting point

The operation proposed at the beginning of this exploration was

for positive real . We called this special division.

A natural symmetric companion subsequently emerged:

which we called special multiplication.

The purpose of the conversation was exploratory: rather than claiming these operations constitute a new mathematical theory, we asked what structures arise if they are taken seriously.

What follows is an audit of the main claims. “Theorem/result” below means that the claim follows directly from the definitions; it does not mean that the result is new to mathematics.

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1. Fundamental identities — VERIFIED

Taking logarithms gives

and

Consequently,

and

Thus # and @ jointly encode the two directional exponentiations and .

Swapping the operands gives

but

Thus # is symmetric whereas @ is multiplicatively antisymmetric.

Self-operation gives

and

With 1,

and

Neither operation is generally associative.

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2. The symmetric/antisymmetric interpretation — VERIFIED

Set

Then

Thus the two operations arise from the standard symmetric/antisymmetric decomposition of the ordered pair

Equivalently, starting from the asymmetric positive-valued operation

one can always form

and

Our # and @ are precisely this construction for .

This is probably the cleanest abstract description of the construction.

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3. The natural coordinate — VERIFIED

Define

Then

and

Equivalently,

and

This is one of the strongest structural observations in the conversation: after normalization, # and @ literally encode the sum and difference of the -coordinates.

We provisionally called a “special logarithm,” but this terminology is invented for the discussion and should not be confused with established uses of “super-logarithm” or other terminology.

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4. Why appears — VERIFIED

Differentiating,

Hence increases on , reaches its unique maximum at , and decreases on .

The maximum is

Therefore

for every positive real , with equality only at .

Similarly,

Thus is intrinsically distinguished by the operation; it was not inserted into the definition.

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5. Special partners — VERIFIED

We have

iff

iff

Thus it is natural to call distinct satisfying this relation special partners.

The familiar example is

Indeed,

All nontrivial positive-real partner pairs can be parameterized as

for .

Equivalently,

As ,

So is the point where the two real partner branches meet.

For every , there is exactly one distinct partner , and conversely.

Among distinct positive integers, the only solution of

is, up to reversal,

Thus really is exceptional in the integer theory.

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6. The folded/two-sheet geometry — VERIFIED, with interpretation caveats

The range of

on the positive reals is

For

the equation

has two positive-real solutions: one in , and one in .

Solving with Lambert ,

The two real Lambert branches give

and

For

these give exactly

Thus the positive-real theory can legitimately be pictured as a two-sheeted structure for , with the sheets meeting at .

For , corresponding to , there is only one positive-real branch.

Calling this a “special number space” is an interpretation rather than a theorem, but the underlying branch structure is real.

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7. @-equivalence and the quotient — VERIFIED

Define

Because this is equivalent to

it is genuinely an equivalence relation.

A natural pseudometric is

It obeys the triangle inequality automatically.

It is only a pseudometric on positive reals because, for example,

After quotienting by , however, it becomes an ordinary metric.

One important caveat discovered later is that # does not descend to this quotient. For example,

but

whereas

and 4 and 16 are not @-equivalent.

So identifying partners throws away information needed by #.

The full two-sheet structure is therefore richer than the quotient.

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8. Geometric identities — VERIFIED

If

then

with

Hence

and

The first is simply the Euclidean sum/difference identity; the second is the corresponding difference-of-squares identity.

Any analogy with Lorentzian geometry or physics should be regarded as suggestive only; no physical interpretation has been established.

Another verified identity is

where defined.

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9. Complex extension — CORRECTION REQUIRED

The earlier discussion said, somewhat too casually, that complexifying

immediately gives infinitely many inverse values

There is a branch subtlety here.

If specifically means the principal complex logarithm, not every Lambert- branch automatically solves the original principal-logarithm equation. Solving

forgets the information about which logarithm branch was used.

The safer statement is:

If logarithm branches are explicitly included, so that

then the complex theory naturally involves Lambert- branches and logarithmic branch bookkeeping.

Thus the idea of an infinite-sheeted Riemann-surface-type extension remains reasonable, but the transcript's first presentation of it was oversimplified.

In particular, the numerical complex values previously quoted for “” solve

but should not automatically be described as solutions of

when is required to be principal.

This is the most important technical correction I found in the audit.

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10. Definition of #-prime

To investigate number theory we proposed:

An integer is #-composite if

for some integers

Otherwise is called a #-prime.

This is a proposed definition, not something forced uniquely by the original operator. Other notions of “special prime” could certainly be studied.

Nevertheless, this definition is the direct analogue of defining ordinary composite numbers via nontrivial multiplication.

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11. Integrality criterion for a#b — VERIFIED

Since

is an integer exactly when is a perfect square.

Writing for the exponent of prime in , this requires

to be even for every prime .

This yields a complete elementary classification.

If are both even,

automatically.

If are both odd,

If exactly one operand is even, then

This claim survives the audit.

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12. Ordinary primes are #-prime — VERIFIED

Suppose an ordinary prime satisfied

with integers .

Then

The prime factorization of the right side implies both and are powers of :

Therefore

so

But and , giving

a contradiction.

Hence

Every ordinary prime is a #-prime

The converse is very false.

For example,

are #-prime despite being ordinarily composite.

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13. #-composites up to 1000 — VERIFIED, WITH ONE COUNTING CORRECTION

An exact integer search confirms that the only #-composites at most 1000 are

Their factorizations include

The previous response said this meant there were 990 #-primes between 2 and 1000. That was an off-by-one counting error.

There are 999 integers from 2 through 1000 inclusive, so the correct count is

#-primes between 2 and 1000 inclusive

The list of ten composites itself was correct.

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14. Special squares — VERIFIED AS A DEFINITION

Since

it is natural to call

a special square.

Thus the sequence begins

rather than

Consequently 9 is a #-prime despite being an ordinary square, whereas 27 is a special square.

Again, “special square” is terminology proposed in the conversation rather than an established concept.

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15. An infinite family of #-composites — VERIFIED

For ,

and therefore

This gives

as an infinite family of #-composites.

Another obvious infinite family is

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16. #-primes have natural density 1 — NOW PROVED

The previous discussion presented this as a conjecture. It can actually be proved very easily.

For integers ,

and

Hence

Therefore, if

then necessarily

so

There are only

integer pairs satisfying this bound.

Every #-composite at most must arise from at least one such pair. Consequently

Therefore

and hence

So the “Special Prime Density Conjecture” in the original conversation should be upgraded to a theorem.

This also explains why #-primes looked overwhelmingly abundant in the first 1000 integers.

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17. Nonunique #-factorisation — VERIFIED, AND STRONGER THAN PREVIOUSLY STATED

We found

This is correct.

Also,

This too is correct.

The previous discussion stopped short of establishing whether the final #-prime leaves could actually differ.

They can.

Because 8 is itself #-prime,

is already a factorization into #-primes:

But

and

so the alternative factorization ultimately has five copies of the #-prime 2 as leaves:

Thus

has genuinely different complete #-prime factorizations.

Similarly, 9 is #-prime, so

has prime leaves

whereas

and

give

Therefore unique factorization into #-primes definitively fails.

This is stronger than what was established in the original thread.

Because # is also nonassociative, a complete factorization is naturally a binary expression tree, not merely an unordered product of primes.

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18. Special divisibility — PROPOSED DEFINITION

A natural definition is

if there exists an integer such that

Unlike ordinary divisibility, there is no reason for this relation to be transitive because # is nonassociative.

It may therefore be more natural to study special divisibility as a graph or hypergraph of factor relationships rather than as the familiar divisibility partial order.

This remains a proposed research direction rather than an established theorem.

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19. Things that should NOT yet be claimed

The exploration does not establish that this is a fundamentally new mathematical structure unknown in the literature.

The symmetric/antisymmetric decomposition itself is standard mathematical machinery, and equations such as

and their Lambert- parameterizations are well studied.

Likewise, inventing terminology such as “special logarithm,” “special square,” and “#-prime” does not establish novelty.

What may be interesting is the particular package obtained by starting from

and systematically treating it together with

especially the resulting integer factorization theory. Whether that package overlaps an existing named theory is precisely the sort of question on which expert feedback would be useful.

Nor should the complex extension be presented as settled; it needs careful treatment of logarithmic and Lambert- branches.

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20. Concise picture of what survived the audit

The central structure is:

They decompose mutual exponentiation:

Their natural coordinate is

in which

and

The function peaks at , producing a real two-sheet structure and special partner relation

The unique unequal positive-integer partner pair is

On the integer side, defining to be #-composite when

produces a sparse set of composites. Every ordinary prime is #-prime, but many ordinary composites are also #-prime.

Indeed,

so

And unique #-prime factorization fails dramatically, for example:

ultimately gives the two different #-prime leaf sets

and

Likewise

gives

versus

So the number-theoretic theory is genuinely non-UFD-like, highly nonassociative, and extremely prime-rich.

Bottom line

After checking the thread, I would characterize it this way:

The main algebraic observations are sound. The real two-sheet/Lambert- picture is sound. The integer integrality criterion and initial computational results are sound. The proposed #-prime theory is internally coherent, #-primes provably have density 1, and unique #-prime factorization genuinely fails.

The main correction is that the complex extension was described too casually: logarithm branches must be tracked explicitly. There was also a minor counting error: there are 989, not 990, #-primes from 2 through 1000 under the proposed definition.

The interesting questions for expert readers are therefore not merely “did ChatGPT make an algebra mistake?” but also:

1. Is this pair of operations, or an equivalent construction, already studied under some established framework?
2. Is there a more natural algebraic setting for # and @ than the one developed here?
3. Is the proposed notion of #-primality mathematically fruitful, or is another definition more natural?
4. What can be said about the graph/tree structure of #-factorizations?
5. Can the number of #-composites up to be estimated much more sharply than ?
6. Which integers possess multiple complete #-prime factorization trees, and how common are they?
7. What is the correct global complex/Riemann-surface formulation once all logarithmic branches are treated rigorously?

Those seem to me like good places for human mathematicians to take over.