x^n + y^n = z^n has no solution if n>2.

Proof:

• n=4 [by Fermat]
• n=3 [by Euler]
• n=5 [by Legendre]
• n=12 [by Dirichlet]
• n=7 [by Lamé]
• odd primes n<=100 (in fact 84 but let's imagine we are in the dozenal world) [by Sophie Germain]
• odd primes n such that 2*n+1 is also prime (if none of x, y, z is divisible by n) [by Sophie Germain]
• odd primes n such that there is a prime of the form k*n+1 where k<=14 and k ends with 2, 4, 8, or X (i.e. k is divisible by 2 but not by 3) (if none of x, y, z is divisible by n) [by Sophie Germain]
• all Bernoulli regular primes n (the Bernoulli irregular primes below 1000 are 31, 4E, 57, 85, 87, XE, 105, 111, 175, 195, 19E, 1X7, 1E7, 205, 217, 21E, 24E, 255, 277, 285, 295, 2X1, 2E1, 301, 325, 327, 32E, 34E, 377, 391, 397, 3X5, 401, 40E, 415, 427, 431, 435, 437, 447, 45E, 465, 46E, 481, 485, 48E, 497, 507, 527, 531, 535, 545, 565, 575, 577, 585, 58E, 59E, 611, 615, 61E, 655, 675, 68E, 745, 76E, 791, 7X1, 7EE, 801, 835, 841, 855, 865, 871, 8X7, 8XE, 8E7, 901, 905, 90E, 91E, 927, 95E, 971, 995, 9E1, 9EE, X37, X4E, X6E, X9E, E11, E21, E25, E2E, E31, E45, E67, E71, EE5) (if none of x, y, z is divisible by n) (it is known that there are infinitely many Bernoulli irregular primes end with 7 or E (i.e. = 3 mod 4)) (it is excepted that about 73.41% (the exact value is e-1/2, where e = 2.875236069822... is the base of the natural logarithm) odd primes are Bernoulli regular) [by Kummer]
• all Euler regular primes n (the Euler irregular primes below 1000 are 17, 27, 37, 3E, 51, 57, 5E, 67, 85, E5, E7, 105, 141, 167, 181, 18E, 19E, 1E1, 217, 21E, 251, 255, 25E, 271, 277, 2XE, 301, 325, 327, 34E, 365, 391, 3XE, 3E7, 401, 40E, 437, 485, 497, 4E1, 517, 527, 535, 541, 545, 577, 585, 611, 61E, 637, 655, 665, 687, 68E, 69E, 705, 70E, 71E, 727, 735, 737, 751, 7EE, 80E, 82E, 85E, 865, 867, 8X5, 8X7, 8XE, 8E7, 90E, 91E, 955, 971, 987, 995, 9X7, 9XE, 9E1, 9EE, X07, X11, X6E, X77, X9E, XEE, E15, E31, E45, E67, E91, E95, EE7) (if none of x, y, z is divisible by n) (it is excepted that about 73.41% (the exact value is e-1/2, where e = 2.875236069822... is the base of the natural logarithm) of the odd primes are Euler regular) [by Vandiver]
• all odd primes n<=60000 (in fact 60408 but let's imagine we are in the dozenal world) [by Samuel Wagstaff]
• all odd primes n such that 2^(n-1) ≠ 1 (mod n^2) (there are only two known exceptions: 771 and 2047) [by Wieferich]
• all odd primes n such that 3^(n-1) ≠ 1 (mod n^2) (there are only two known exceptions: E and 406217) [by Mirimanoff]
• all odd primes n such that b^(n-1) ≠ 1 (mod n^2) for some prime b≤20 and b ends with 5 or E (i.e. b = 5 mod 6) (in fact only b = 5, E, and 15 (not include b = 1E) but let's imagine we are in the dozenal world) [b=5 by Vandiver, b=E and b=15 by Frobenius]
• all odd primes n end with 5 or E (i.e. n = 5 mod 6) such that b^(n-1) ≠ 1 (mod n^2) for some prime b≤20 and b ends with 1 or 7 (i.e. b = 1 mod 6) (note that these b are 7, 11 and 17, and the product of them is 1001, the famous Hardy-Ramanujan number) [by Frobenius]
Known odd primes n such that b^(n-1) = 1 (mod n^2) for primes b≤20
b known odd primes n such that b^(n-1) = 1 (mod n^2)
2 771, 2047
3 E, 406217
5 1002E, 1E51E, 15XX7E61, 39E028717, 1369315921, 306E7457755
7 5, 1E854E
E 5E
11 5EE, 703407
15 3, 22771, 243XE, 7882491569E
17 3, 7, 11, 37, E5, 19151E01
1E 11, 9E8251, 470X031, 301X56181E, 35214X7XX155

(this list only consider odd primes n, since for the only even prime n=2, the equation x^n + y^n = z^n has a solution (in fact, infinitely many solutions), however, of course n=2 satisfies b^(n-1) = 1 (mod n^2) for prime b ends with 1 or 5 (i.e. b = 1 mod 4) and not for prime b ends with 7 or E (i.e. b = 3 mod 4), also not for primes b = 2 and b = 3)

• all odd primes n such that k^(n-1) != 1 (mod n^2) for some prime k<=100 (in fact 95 but let's imagine we are in the dozenal world) [by Frobenius]
• all odd primes n [by Wiles]