Project Euler Problem 753

Fermat's Last Theorem states that no three positive integers a, b, c satisfy the equation for any integer value of n gre

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Project Euler Problem 753

Solution

Answer: 4714126766770661630

Using the cubic-residue structure of $\mathbb{F}_p$ and the CM theory of the elliptic curve

$$x^3+y^3=1,$$

one can derive:

  • If $p \equiv 2 \pmod 3$, the cube map is a bijection, so

$$F(p)=(p-1)(p-2).$$

  • If $p \equiv 1 \pmod 3$, write

$$4p=A^2+27B^2$$

with $A\equiv 1\pmod 3$. Then

$$F(p)=(p-1)(p-8+A).$$

  • Special case:

$$F(3)=2.$$

This matches the examples:

  • $F(5)=12$,
  • $F(7)=0$.

Summing over all primes $p<6{,}000{,}000$ gives:

Answer: 4714126766770661630