Mersenne numbers
 

Am I in the correct place?


Let be p and q two arbitrary Mersenne numbers.


I want a proof that p*q-1 can never be a square.


p*q-1 instead can be a power of 3 as in the cases


3*3-1, 7*31-1 and 127*63-1...in these last cases p*q+1 is an even semi-prime


3*3+1=5*2
7*31+1=109*2
127*63+1=4001*2


Do you believe that another example can be found of Mersenne numbers p and q such that p*q-1 is a power of 3? I don't believe

[QUOTE=enzocreti;515717]Am I in the correct place?


Let be p and q two arbitrary Mersenne numbers.


I want a proof that p*q-1 can never be a square.[/QUOTE]
You got it. If n is the square of a positive integer, then n is a quadratic residue modulo any prime not dividing n.

If p is a Mersenne number, then p == 3 (mod 4). Therefore, p has at least one prime factor l == 3 (mod 4). [Of course, l = p if p happens to be prime.] Now,

l == 3 (mod 4), so -1 is a quadratic non-residue (mod l).

If n = p*q - 1, then n == -1 (mod l), so n is not a quadratic residue (mod l).

Therefore, p*q - 1 is not a square.

Oeis sequence
 

1, [B]8, 216, 8000[/B], 343000, 16003008, 788889024, 40424237568, 2131746903000, 114933031928000, 6306605327953216, 351047164190381568, 19774031697705428416, 1125058699232216000000, 64561313052442296000000 [SIZE=-1]([URL="https://oeis.org/A002897/list"]list[/URL]; [URL="https://oeis.org/A002897/graph"]graph[/URL]; [URL="https://oeis.org/search?q=A002897+-id:A002897"]refs[/URL]; [URL="https://oeis.org/A002897/listen"]listen[/URL]; [URL="https://oeis.org/history?seq=A002897"]history[/URL]; [URL="https://oeis.org/search?q=id:A002897&fmt=text"]text[/URL]; [URL="https://oeis.org/A002897/internal"]internal format[/URL]) [/SIZE][URL="https://oeis.org/A002897"]A002897[/URL]
a(n) = C(2*n,n)^3.
[SIZE=-1](Formerly M4580 N1952)


I dont know what is this sequence and the C function but 8,216 and 8000 are

8=3*3-1
216=7*31-1
8000=63*127-1

so exactly the product of the Mersenne numbers minus one that are a 3rd power.

[/SIZE]
[SIZE=-2] +30
27 [/SIZE]

7*31+4 and 127*63+4
 

interesting is that 7*31+4 has 17=4^2+1 as greatest prime factor and 127*63+4 has 1601=40^2+1 as greatest prime factor