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By Fokkinga M.M.

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In fact, it su ces to show that one of the factors is both 1 and n3 , 1. Note that for n = 2 the expression n3 , 1 has value 7, which is prime, so we'd better not try to prove that this expression is never prime. For n 2 certainly n , 1 2 , 1 = 1. This is one comparison. On the other hand, also for n 2, n,1 22n,1 nnn,1 Thus, 0 n , 1 m3 , 1 if n 2. This shows that n3 , 1 is never prime for n 2. One special algebraic form for numbers, which was historically of recreational interest, but is now also of practical interest, is 2n , 1.

Next, 3  34 + 1 = 103 is prime, but 131071  103 = 55. Next, 4  34 + 1 = 137 is prime, but 131071  137 = 99. Next, 5  34 + 1 = 171, which is divisible by 3. Next, 6  34 + 1 = 205,pvisibly divisible by 5. Next, 7  34 = 1 = 239, which is prime testing prime divisors 2; 3; 5; 7; 11; 13 all  239 16. But 131071  239 = 99. Next,p8  34 = 1 = 273, which is divisible by 3. Next, 9  34 = 1 = 307, which is prime testing prime divisors  307 18. But 131071  307 = 289. Next, 19  34 + 1 = 341, which is divisible by 11.

If 2089 were not prime, then it would have a prime factor  2089 46, but also 1 mod 58. There aren't any such things, so 2089 is prime. 4 Factoring 3n - 1 We continue with more examples using Fermat's observation about factors of special numbers of the form bn , 1. Every number 3n , 1 for n 1 has the obvious factor 3 , 1, so is not prime. But this is a rather weak statement, since we might want the whole prime factorization, or at least be curious whether or not 3n , 1=3 , 1 is prime. Fermat's trick is helpful in investigating this, in the same way that it was helpful in looking at Mersenne numbers.

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A gentle introduction to category theory. The calculational approach by Fokkinga M.M.


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