1.Introduction

I show two properties about Mersenne number.
Euler showed the following theorem.

Theorem(Euler)

Let p = 3 (mod 4) be prime and q=2p+1 is also prime.
Then q divides Mersenne number M(p)=2^p-1.

The theorem of Euler is related to Quadratic residue.

I searched the similar theorem that is related to other higher power residue,
and found two theorems.

One is related to sextic residue,and one more is related to octic residue.

In the same way as Euler,we can prove the theorems.


2. Theorem
          

Theorem 1.

Let p is prime and q=6p+1 is also prime.
If q=27x^2+y^2 and q=7 mod 8,q divides Mersenne number M(p)=2^p-1.

Since q=27x^2+y^2 and q=7 mod 8,then
z^6 = 2 mod q has a integer solution z.(By cubic and quadratic reciprocity law)

So,2^p = 2^{(q-1)/6} = z^(q-1) = 1 mod q.
Therefore,q=6p+1 divides Mersenne number M(p)=2^p-1.

Example: p < 1000
    p    q
   37  223 = 27*  1^2 + 14^2 , 2^37  - 1=0 mod   223
   73  439 = 27*  3^2 + 14^2 , 2^73  - 1=0 mod   439
  233 1399 = 27*  3^2 + 34^2 , 2^233 - 1=0 mod  1399
  397 2383 = 27*  9^2 + 14^2 , 2^397 - 1=0 mod  2383
  461 2767 = 27*  7^2 + 38^2 , 2^461 - 1=0 mod  2767
  557 3343 = 27*  9^2 + 34^2 , 2^557 - 1=0 mod  3343
  577 3463 = 27* 11^2 + 14^2 , 2^577 - 1=0 mod  3463
  601 3607 = 27*  3^2 + 58^2 , 2^601 - 1=0 mod  3607
  761 4567 = 27* 13^2 +  2^2 , 2^761 - 1=0 mod  4567



Theorem 2.

Let p is prime and q=8p+1 is also prime.
If q=64x^2+y^2 for some odd x,q divides Mersenne number M(p)=2^p-1.

Since q=64x^2+y^2 for some odd x,then
z^8 = 2 mod q has a integer solution z.(By octic reciprocity law)

So,2^p = 2^{(q-1)/8} = z^(q-1) = 1 mod q.
Therefore,q=8p+1 divides Mersenne number M(p)=2^p-1.

Example: p < 1000
    p    q
   11   89 = 64*  1^2 +  5^2 , 2^11  - 1 = 0 mod    89
   29  233 = 64*  1^2 + 13^2 , 2^29  - 1 = 0 mod   233
  179 1433 = 64*  1^2 + 37^2 , 2^179 - 1 = 0 mod  1433
  239 1913 = 64*  1^2 + 43^2 , 2^239 - 1 = 0 mod  1913
  431 3449 = 64*  5^2 + 43^2 , 2^431 - 1 = 0 mod  3449
  761 6089 = 64*  5^2 + 67^2 , 2^761 - 1 = 0 mod  6089
  941 7529 = 64*  5^2 + 77^2 , 2^941 - 1 = 0 mod  7529
  857 6857 = 64*  7^2 + 61^2 , 2^857 - 1 = 0 mod  6857