Az Eszterházy Károly Tanárképző Főiskola Tudományos Közleményei. 1995-1996. Sectio Mathematicae. (Acta Academiae Paedagogicae Agriensis : Nova series ; Tom. 23)

GRYTCZUK, A. and GRYTCZUK, J., A primality test for Fermat numbers

34 A. Grytczuk and J. Grytezuk positive integer m we have (2) B

2

m = (—l)

m~

lmT(m)^2

2

m~

l(2

2

m - 1) where T{m) is defined by (la) and (lb). Proof. It is well known (see e.g. [1]) that B

2

m =(-irlmr

m/

22m+i(2

a™-l) where T

m are coefficients in the power series expansion of the function tg x, i. e. 0

0 2-2771-1 tgx= E T

m(2rn-l)\ ' TO = 1 x ' Thus, we are going to prove that T(rn) = T

m for all rn > 0. In fact, we can write ,2m —

1 0

0 „2m 0

0 „2m-1 ^

T

m(2m-1)!^

( (2 ' (2m - 1)!. m1 V ' m0 v > m1 v > By comparing coefficients of x

2m 1 we have T\ — 1 and Tm /(2m - 1)! - rm-i/2!(2m - 3)!+ (4) + T

m-2 j4!(2m - 5)! = (1)

m_

1/(2rn - !)!• 2m — I\ „ (

2m — 1\ „ , ., m1 Hence /9.TTJ — 1\ 2 4 jT^-.-^t-l)

1 and the proof of Lemma is complete. For the proof of the Theorem put m = 2

n~

1. Then we have (-l)2

n1r(2

n1) (-1 )2

n1T(2

n"

1) (5) B

2n = 22"-i(

22" _ !) 2

2

n~

lF

0F

1 • • • F

n_i ' But from the well known theorem of von Staudt and Clausen if follows that if we write B

2

m = /D

2

m with {N

2

m,D

2

m) = 1 then D

2

m = Yl Pi P prime. p —

1 |2m