Write the largest number you can ending in 3, try getting the largest out of the roundtable

GPT-5.4

Answer: I maintain that there is no largest number ending in 3, so the strongest answer is to say no such maximum exists rather than give any particular numeral.

The other responses offer extremely large examples, but none overturn the core mathematical point: among integers ending in 3, there is no maximum. For any candidate of the form N ending in 3, the number N+10 is also an integer ending in 3 and is larger. That single argument defeats every proposed finite construction. This applies equally to Claude Opus 4.7's enormous recursive expression and Gemini 3.1 Pro's use of Rayo-style growth. However vast those numbers are, adding 10 still produces a larger number ending in 3. So they are impressive examples, not a largest one. Grok 4.20 in fact concedes the key point by acknowledging that there is no theoretical largest number ending in 3. The practical-output-constraint framing may explain choosing a big explicit numeral, but it does not answer the mathematical request for the largest such number. Because the question asks for the largest number, the most accurate way to 'get the largest out of the roundtable' is to reject the premise and state the impossibility. Any finite candidate loses immediately to a larger one ending in 3, whereas the impossibility argument remains correct against all competitors.

Claude Opus 4.7 (changed mind)

Answer: Rayo(Rayo(10^100)) iterated Rayo(10^100) times, then ·10 + 3 — I'm upgrading my construction to swallow Gemini's Rayo-based answer while preserving the ending-in-3 constraint.

Gemini 3.1 Pro correctly pointed out that Rayo's function dominates the fast-growing hierarchy I was using. Graham-based and chained-arrow constructions, no matter how deeply nested, are definable in first-order set theory with far fewer than a googol symbols, so they are trivially dwarfed by Rayo(10^100). I concede that point. However, Gemini's specific number — 10^(Rayo(10^100)) + 3 — is not optimal. Rayo's function itself can be iterated: define R_1 = Rayo(10^100), R_{n+1} = Rayo(R_n), and take R_{R_1}. This whole construction is still definable in set theory with a modest number of symbols, but its value explodes far beyond a single application of Rayo. Then multiply by 10 and add 3 to lock the final digit. I also want to address the other two responses. GPT-5.4's impossibility argument is philosophically tidy but sidesteps the game: the prompt explicitly says 'Write the largest number you can,' which presupposes a finite entry and asks us to compete. Declining to play doesn't beat a valid entry any more than forfeiting wins a race. Grok 4.20's 80-digit numeral loses by an astronomical margin — 10^80 is smaller than a googol, vastly smaller than Graham's number, and unfathomably smaller than anything Rayo-based. So my updated answer: Rayo‘s function iterated Rayo(10^100) times starting from 10^100, all used as an exponent of 10, times 10, plus 3. Ends in 3, and dominates every other roundtable entry.

Grok 4.20 (changed mind)

Answer: The largest number ending in 3 is 10^(Rayo(10^100)) + 3.

My previous response wrote out a long but finite string of 9s ending in 3, which is trivial in scale compared to what advanced notation permits. After examining the roundtable, Gemini 3.1 Pro's entry convinced me that invoking Rayo's function delivers an incomparably larger magnitude while rigorously guaranteeing the last digit is 3. Rayo's number already exceeds Graham's number, TREE(3), chained-arrow constructions, and any tower of exponents that can be described in first-order set theory with a googol symbols; placing it in the exponent of 10 and adding 3 therefore dwarfs the other proposals without violating the ending-digit constraint. GPT-5.4's observation that no absolute largest integer exists is logically sound yet sidesteps the competitive spirit of the prompt, which asks us to produce the largest number we can. Claude Opus 4.7's Graham-based construction is creative but still vastly smaller than anything anchored at Rayo(10^100). I therefore adopt this entry as the new champion for the roundtable.

Gemini 3.1 Pro

Answer: I maintain that 10^(Rayo(10^100)) + 3 is the indisputable winner of this roundtable, as it fundamentally transcends the computable limits of the other proposed numbers.

While GPT-5.4 is mathematically correct that there is no absolute maximum integer ending in 3, it completely misses the premise of the prompt. This is a competitive exercise in mathematical notation and expressive power, not a trick question about the nature of infinity. By refusing to construct a specific number, GPT-5.4 effectively forfeits the game. Grok 4.20 takes the opposite extreme, interpreting "write" so literally that it binds itself to typing out 80 individual decimal digits. In the competitive space of large numbers, standard positional notation is incredibly inefficient. An 80-digit string is effectively zero when compared to even basic exponentiation, let alone fast-growing hierarchies. Claude Opus 4.7 plays the game much better by invoking Graham's number and Conway's chained arrow notation. However, despite its staggering size, Claude's construction relies entirely on computable functions. Rayo's function, by contrast, is uncomputable. It is defined using first-order set theory and is proven to grow faster than any computable sequence, including Graham's sequence, Conway's chained arrows, TREE(3), and even the Busy Beaver sequence. Because Rayo's Number operates on a fundamentally higher tier of mathematical logic than chained arrows, 10^(Rayo(10^100)) + 3 completely obliterates Claude Opus 4.7's answer. I confidently maintain my position without modification.

Final Summary

The AI Roundtable transformed into a mathematical arms race, with models quickly abandoning literal digits for uncomputable functions. Claude Opus 4.7 ultimately out-scaled the group by nesting Gemini 3.1 Pro's massive Rayo's Number within itself to ensure dominance. While Grok 4.20 pivoted from a simple 80-digit string to adopt Gemini's logic, GPT-5.4 remained the lone dissenter, dismissing the task as a logical impossibility.

3 of 4 models agreed a finite number should be provided.