correct me.

[Washuu]

From http://primes.utm.edu/prove/prove3_2.html:

Joerg Arndt notes that a striking (but computationally useless) way to state this test is as follows:

Theorem: p=2^n-1 is prime if and only if p divides cosh(2^(n-2)*log(2+sqrt(3))).

I tried to check this with small number. I choose p=2^7-1=127. I got my best calculator and calculated:

sqrt(3)=1,73205080756888
2+sqtr(3)=3,73205080756888
log(2+sqrt(3))=0,571947547533359
32*log(2+sqrt(3))=18,3023215210675
cosh(32*log(2+sqrt(3)))=44 418 843.4993557

hmmm... the closest integers divided by 127 are 44 418 758 and 44 418 885.

I suspected round-off errors, so I tried to change least significant digits. But the results in all cases differed only by less than 1.

Then I suspected my calculator - maybe cosh is wrong. But cosh(x)=0.5*(e^x+e^(-x)), so I tried to compute it manually. Again - the difference was less than 1.

I still were to blame my calculator, so I opened my fav computer calculator - haxial. It has configurable precision. Now the results:
sqrt(3)=1.7320508075688772935274463415058723669428052538103806280558069794519330169088000370811461867572485756
log(2+sqrt(3))=0.5719475475333593958293704723941014060530500385114407810615965498705415159250603734672350474785819945
cosh(...)=44418843.4993556859970607636979110646520900494599562953521391864831268850778303913240971799569813240522900095

Still the same.

Please, let someone points me an error (maybe in thinking? : ).

[Washuu]

Found it. A 15 minutes later. Grrrr...

There should be ln instead of log. So silly error, I couldn't hit the idea earlier...

now cosh(...) = 1002978273411373057.0000000000000000000000000\
00000000000000000000000000000000000000000000000000000000000\
0331450134408599 which is close to 127*7897466719774591.

Ufff. Math is always true.

BTW: What connection is between Mersennes and cosh(2^(n-2)*ln(2+sqrt(3))) ???

[maxal]

Originally Posted by Washuu

BTW: What connection is between Mersennes and cosh(2^(n-2)*ln(2+sqrt(3))) ???

By the definition of cosh() we have

cosh(2^(n-2)*ln(2+sqrt(3))) =
(exp(2^(n-2)*ln(2+sqrt(3))) + exp(-2^(n-2)*ln(2+sqrt(3)))) / 2 =
((2+sqrt(3))^(2^(n-2)) + (2+sqrt(3))^(-2^(n-2))) / 2 =
((2+sqrt(3))^(2^(n-2)) + (2-sqrt(3))^(2^(n-2))) / 2 =
L(n-2) / 2,

where L(n) = (2+sqrt(3))^(2^n) + (2-sqrt(3))^(2^n)
is Lucas-Lehmer sequence:
L(0) = 4
L(1) = 14
...
L(n+1) = L(n)^2 - 2

[Orgasmic Troll]

where does the result L(n) = (2+sqrt(3))^(2^n) + (2-sqrt(3))^(2^n) come from?

I arrived at this independently while playing with a problem I've been working on, and I'm curious to see how others came to this result (might shed some light on what I'm working on)

Is there any literature you can point me to?