Prime time for a number theory question

Number Theory Level 3

Let p be a prime number, and let n be a positive integer. How many possible solutions are there to: $n^{3} = p + 1$ ?

The answer is 1.

1 solution

Samuel Sturge
Jul 5, 2019

$n^3 = p + 1$
$n^3 - 1 = p$
$(n^2 + n + 1)(n - 1) = p$
Since p has 2 factors, 1 and p, either $n^2 + n + 1 = 1$ or $n - 1 = 1$
If $n^2 + n + 1 = 1$ , then $n(n + 1) = 0$ , giving (\n = 0,-1. )
Neither of these are positive integers, so we are left with $n - 1 = 1$ , which only has 1 solution : $2$

Nice solution but be careful - you're missing one key point! What happens if you try the same approach to find the number of solutions to $n^{11}=p+1$ ? (Same conditions - $n$ a positive integer and $p$ prime.)

Chris Lewis - 1 year, 11 months ago

Prime time for a number theory question

The answer is 1.

1 solution

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