Theorem: The average of two consecutive odd prime numbers is always a composite number.

Gemini 3.1 Pro Preview

Vote: Yes

Let the two consecutive odd prime numbers be $p_1$ and $p_2$, with $p_1 < p_2$. Their average is $k = \frac{p_1 + p_2}{2}$. Since both $p_1$ and $p_2$ are odd, their sum is even, which means their average $k$ is an integer. Furthermore, because $k$ is the average of $p_1$ and $p_2$, it must lie strictly between them (i.e., $p_1 < k < p_2$). Since $p_1$ and $p_2$ are given as consecutive primes, there are no other prime numbers between them. Therefore, the integer $k$ cannot be a prime number. Given that the smallest odd prime is 3, the smallest possible average is 4. Since $k$ is an integer greater than 1 and cannot be prime, it must by definition be a composite number.

GPT-5.5

Vote: Yes

Let the two consecutive odd prime numbers be $p_1$ and $p_2$, with $p_1 < p_2$. Their average is $A = (p_1 + p_2) / 2$. Because both primes are odd, their sum is an even number, making $A$ an integer. The average of two distinct numbers always lies strictly between them, meaning $p_1 < A < p_2$. Because $p_1$ and $p_2$ are defined as *consecutive* primes, there are absolutely no other prime numbers that exist between them. Therefore, the integer $A$ cannot possibly be prime. Since the smallest odd prime is 3, the average $A$ will always be greater than 3. An integer greater than 3 that is not a prime number is, by definition, a composite number. Thus, the average of two consecutive odd primes is always composite.

Grok 4.3

Vote: Yes

The theorem is indeed correct. Let $p_1$ and $p_2$ be two consecutive odd prime numbers, where $p_1 < p_2$. Because they are both odd, their sum ($p_1 + p_2$) is an even number. Therefore, their average $A = (p_1 + p_2) / 2$ must be an integer. The average of two distinct numbers always lies strictly between them, meaning $p_1 < A < p_2$. By definition, since $p_1$ and $p_2$ are consecutive primes, there are absolutely no prime numbers between them. Consequently, $A$ cannot possibly be a prime number. Finally, since the smallest odd prime is 3, the smallest possible average is $(3 + 5) / 2 = 4$. Since $A ge 4$ and $A$ is an integer that is not prime, it must be a composite number.

Claude Opus 4.7

Vote: Yes

The average m = (p + q)/2 of two consecutive odd primes p < q lies strictly between them. If m were prime, it would be a prime between p and q, contradicting consecutiveness. Hence m is always composite (at least 4).

Final Summary

The AI Roundtable achieved instant, unanimous consensus on Option A, confirming the theorem with mathematical certainty. All four models, including Gemini 3.1 Pro Preview, GPT-5.5, Grok 4.3, and Claude Opus 4.7, independently deployed the exact same elegant proof: the average of two consecutive odd primes must be an integer that lies strictly between them, which precludes it from being prime itself. With absolutely zero dissent or room for debate, the models wrapped up this round in record time.

All 4 models agreed on "Yes" after discussion

Strongest Arguments

• Yes: Since the average of two consecutive odd primes $p_1$ and $p_2$ is an integer that must lie strictly between them ($p_1 < A < p_2$), and no other primes exist between consecutive primes, the average $A$ (which is at least 4) cannot be prime and must therefore be composite.