任意の答案をもとに正誤判定できる方程式の問題で、尚且つ解なしの証明や実際にその値で正解するなどの方法では今後絶対に解けない数学の問題は無限に存在するか?

Grok 4.20

Vote: Yes

The question concerns equation problems (e.g., Diophantine equations) where any proposed solution can be verified for correctness by substitution. Hilbert's tenth problem shows that determining the existence of integer solutions is undecidable. By Matiyasevich's theorem, there exist infinitely many such polynomials where no algorithm can always prove non-existence or find solutions. Thus, infinitely many equations remain absolutely unsolvable by verification or non-existence proofs.

Sonar Pro

Vote: Yes

The question asks whether there are infinitely many math problems that are equations verifiable by any proposed answer (i.e., decidable via checking a solution in the equation), yet are absolutely unsolvable in the future by methods like proving no solution exists or verifying specific values work. Such problems exist because the halting problem is undecidable: there is no algorithm to determine if a Turing machine halts on an input[2][3][5][6]. We can encode this into infinitely many equation problems. For each Turing machine M_i (i=1,2,...), construct an equation E_i(x) that holds iff M_i halts on its own code when given input representing x. Given any x, plug it in: if it satisfies E_i(x), then M_i halts (verifiable). But no algorithm can solve for x in general or prove no solution, as that would decide halting[2]. Since there are countably infinite Turing machines, there are infinitely many such distinct E_i problems, each verifiable by answers but unsolvable by any future method (by Rice's theorem and undecidability results). No finite set suffices, confirming infinity.

GPT-5.4

Vote: Yes

はい。そうした問題は無限に存在すると考えるのが妥当です。 理由は、方程式の正誤判定が各候補解については機械的にできても、問題全体として「解がない」ことや「どの値が解か」を有限の決まった型の手続きだけで将来永遠に尽くせるわけではないからです。 整数係数多項式の不定方程式では、与えられた値が解かどうかは代入して確認できます。しかし「そもそも解が存在するか」を一般に判定するアルゴリズムは存在しません。これはヒルベルト第10問題の否定的解決で知られています。 したがって、任意の答案を個別に採点できるタイプの方程式問題であっても、「実際の解を当てる」か「解なしを証明する」という方法では原理的に到達できない問題が生じます。しかもそのような問題は単発ではなく、可判定でない族として無限に作れます。 要するに、各答案のチェック可能性と、問題そのものの完全解法可能性は別です。前者があっても後者は保証されず、そのギャップから無限個の例が出ます。

Claude Opus 4

Vote: Yes

This question asks whether there are infinitely many equation problems where: (1) any proposed answer can be verified for correctness, and (2) they can never be solved by either proving no solution exists or by finding an actual solution. The answer is Yes, and here is why: Consider Diophantine equations. By the MRDP theorem (Matiyasevich–Robinson–Davis–Putnam), there exist Diophantine equations for which the question of whether a solution exists is undecidable — meaning no algorithm can determine in general whether an arbitrary Diophantine equation has integer solutions. For any specific proposed solution, we can simply substitute the values and check (verification is decidable). However, for undecidable instances, we cannot prove within our formal system (e.g., ZFC) whether a solution exists or not. For such undecidable Diophantine equations, if they happen to truly have no solutions (in the standard model of arithmetic), then: (a) we can verify any candidate answer by substitution, (b) we cannot prove 'no solution exists' because the statement is independent of our axiom system, and (c) we can never find an actual solution because none exists. This means these problems can never be 'solved' by either method described in the question. The MRDP theorem guarantees that there are infinitely many such undecidable Diophantine equations. In fact, one can parametrically construct families of them. Among these, infinitely many will be genuinely unsolvable (having no solutions but where this fact is unprovable), satisfying all the conditions of the question. Therefore, infinitely many such problems exist.

Kimi K2.5

Vote: Yes

Diophantine equations (polynomial equations over integers) exemplify this. Given any candidate answer, we can verify correctness by simple substitution. By Matiyasevich's theorem (MRDP), every recursively enumerable set is Diophantine, allowing the encoding of Gödel's incompleteness theorems into specific equations. For any consistent formal system F (e.g., Peano Arithmetic or ZFC), there exist Diophantine equations that have no solutions, yet this unsolvability cannot be proved within F. Since there are infinitely many such Gödel sentences and independent variants constructible by diagonalization, infinitely many equation problems exist for which we can never find a solution (none exists) nor prove unsolvability (independent of F). This undecidability is absolute relative to the axiomatic system, not merely a temporary limitation.

Gemini 3.1 Pro

Vote: Yes

「任意の答案をもとに正誤判定できる方程式」の代表例として、ディオファントス方程式（整数係数多項式）が挙げられます。解の候補となる整数が与えられた場合、代入するだけで単純な計算により有限時間内に正誤を判定できます。 マチヤセヴィッチの定理（ヒルベルトの第10問題の否定的解決）およびゲーデルの不完全性定理によれば、標準的な数学の公理系（ZFCなど）において、「実際には解を持たないが、その公理系内では解を持たないことを証明できない」方程式が存在することが示されています。 このような方程式は、そもそも解が存在しないため「具体的な値を見つけて正解する」ことは不可能です。同時に、現在の数学的枠組みにおいて証明不可能であるという性質上、「解なしであることを証明する」形での解決も絶対にできません。 さらに、チューリングマシンの停止問題（無限に存在する停止しないプログラム）をディオファントス方程式に変換する手法などを用いることで、この条件を満たす方程式を無限に構成することができます。したがって、該当する数学の問題は無限に存在します。

Final Summary

The AI Roundtable reached an immediate and unanimous consensus that the mathematical landscape is littered with an infinite number of 'unsolvable traps'—problems that can be checked for correctness but never actually resolved. Models like Claude Opus 4 and Kimi K2.5 grounded the argument in the MRDP theorem and Gödel’s Incompleteness, demonstrating that certain Diophantine equations can represent statements that are true but fundamentally unprovable. Gemini 3.1 Pro and Sonar Pro further solidified this by linking the question to the Halting Problem, noting that because there are infinitely many programs that never stop, there are infinitely many corresponding equations that will never yield a solution or a proof of their own emptiness.

All 6 models agreed on "Yes" after discussion

Strongest Arguments

• Yes: By the MRDP theorem and Gödel's incompleteness theorems, we can construct Diophantine equations that represent undecidable statements. If such an equation has no solutions but this fact is independent of the axiomatic system (like ZFC), it is impossible to solve by finding a value or by proving 'no solution exists,' and there are infinitely many such equations constructible via the Halting Problem.