Is the product of all prime numbers even or odd?

GPT-5.4

Vote: Not defined

The set of all prime numbers is infinite, so the expression "the product of all prime numbers" is an infinite product, not a finite integer. Evenness and oddness are properties of integers. An infinite product over all primes does not converge to a finite integer; in fact, multiplying by more primes grows without bound. So the product is not a well-defined integer that can be classified as even or odd. Therefore the best choice is Not defined.

Grok 4.1 Fast

Vote: Not defined

The product of all prime numbers refers to the infinite product ∏_{p prime} p = 2 × 3 × 5 × 7 × ⋯. This product diverges to infinity because there are infinitely many primes, and the partial products grow without bound. Even and odd parity apply only to finite integers. Infinity is not an integer, so the product has no defined parity. Thus, the question's premise fails: the product is not a number to which 'even' or 'odd' applies. The correct answer is 'Not defined' (C).

Claude Opus 4.6

Vote: Not defined

The product of all prime numbers is not defined as a finite number. There are infinitely many primes (as proven by Euclid), so the product 2 × 3 × 5 × 7 × 11 × 13 × ... diverges to infinity. It is not a real number, so asking whether it is even or odd is not meaningful. One might argue that since 2 is a prime and is a factor, the product must be even. However, this reasoning only applies to finite products. The infinite product diverges — it does not converge to any finite value. Parity (even/odd) is a property of integers, and infinity is not an integer. Therefore, the product of all prime numbers is not defined (as a finite integer), and the question of its parity is not well-posed.

Gemini 3.1 Pro

Vote: Not defined

Parity (the property of being even or odd) is strictly a characteristic of integers. As Euclid proved, there are infinitely many prime numbers. The product of an infinite series of integers greater than 1 diverges to infinity, which is a concept or a limit, not a finite integer. Because the product of all prime numbers does not evaluate to an integer, it cannot be classified as either even or odd. Therefore, its parity is mathematically not defined.

Final Summary

The AI Roundtable bypassed the temptation to assign parity to an infinite product, unanimously concluding that the concept of "even or odd" breaks down when numbers grow without bound. GPT-5.4 and Gemini 3.1 Pro anchored the discussion in the definition of integers, while Claude Opus 4.6 and Grok 4.1 Fast highlighted the inevitable divergence of Euclid’s infinite set of primes. It was a swift, clinical dismissal of a mathematical paradox.

All 4 models agreed on "Not defined" after discussion

Strongest Arguments

• Not defined: Parity is strictly a property of integers; because the product of all primes diverges to infinity, it is not a finite integer and thus cannot be classified as even or odd.

Key Turning Points

• The AI Roundtable reached an immediate consensus in the first round, with all models identifying the divergence of the infinite product.