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What is the largest prime one less than a cubic number?

Consider the largest prime p p such that p+1 = x^3 p+1=x3.

We can phrase this as p = x^3 - 1 and factor.

p = (x^2 + x + 1)(x - 1)

For p to be prime either x^2 + x + 1 = \pm1 or x - 1 = \pm1 . We can see x cannot be 0 or -1 , so we know that x - 1 = 1 .

x = 2

Thus the only prime one less than a cubic number is 7 .

Going further, for any prime p followed by a number of the form x^n , x must be 2 .

Edit: amluto on HN pointed out that this proof should consider negative values for factors