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 = 2Thus 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