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1

Consider that $6k-1,6k+1$ are both prime exactly when there exists no positive integer $a$ such that $6a\pm 1\mid 6k-1, 6a\pm 1\mid 6k+1$. Since the congruence classes which are relatively prime to the modulus $6$ are $\pm 1$, the only possible numbers which could divide $6k\pm1$ are numbers of the form $6a\pm 1$, and therefore the above divisibility tests ...

5

It depends really on what you mean by open problem in category theory. By that I mean that one must make a clear distinction between research that employs categorical language, and research of category theory. The reason is very simple: category theory is a very widely used language for discussing mathematics. This is much like the situation with set theory. ...

1

The brute force method is effective in this case. Suppose I claim 37998938 is a counterexample to Goldbach. As it happens, the lowest prime that splits 37998938 is 1039. This is the smallest Goldbach-unfriendly number for all primes under 1000. 3325581707333960528 needs the prime 9781 for a Goldbach split. What is the smallest Goldbach-unfriendly number ...

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