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Dirichlet's Theorem: there are infinitely many primes in every arithmetic progression $\{a + nd: n \geq 0\}$ for coprime $a, d$. Consider just the case where $a, d$ can take on prime values or $0$.

Then define $a \oplus a = 0$, $a \oplus 0 = 0 \oplus a = a$ and $a \oplus b$ for $a \neq b$ two prime numbers to be the smallest number in the above progression ($a + nb$) greater than $a$ that is a prime.

For instance $3 \oplus 5 = 3 + 2\cdot 5 = 13$

By Dirichlet's theorem there are infinitely many primes, so choose the smallest one greater than $a$!

Associativity:

$$ (3 \oplus 5) \oplus 7 = 13 \oplus 7 = 13 + 4 \cdot 7 = 41 \\ 3 \oplus (5 \oplus 7) = 3 \oplus 19 = 41 $$

Does associativity hold at least when $a, b, c$ are distinct?

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The prime-triple $(5,7,11)$ gives already a counterexample.

If $f(p,q)$ denotes the above operation, we have $f(5,7)=19$ $f(19,11)=41$ , $f(7,11)=29$ and $f(5,29)=179$

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Take $7\oplus 11=29$, so that $(7\oplus 11)\oplus 13=29\oplus 13=107$. On the other hand we have $11\oplus 13=37$, and $7\oplus(11\oplus 13)=7\oplus 37=229$. So $$ (7\oplus 11)\oplus 13=107\neq 229=7\oplus(11\oplus 13) $$

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