Az Eszterházy Károly Tanárképző Főiskola Tudományos Közleményei. 1994. Sectio Mathematicae. (Acta Academiae Paedagogicae Agriensis : Nova series ; Tom. 22)
GRYTCZUK, J., On Perron's proof of Fermât's two square theorem
On Perron's proof of Fermât's two square theorem JAROSLAW GRYTCZUK Abstract. Fermât's two square theorem states that every prime of the form 4772+ 1 is the sum of two squares. In this note we give a new proof for this using continued fraction expansion of squareroot of non-square integers. It is well known that if a natural number d is not a perfect square the simple continued fraction expansion oîVd is periodic and has the form y/d =< ao, ai, 02, • • • as >j where a l = [(RRII -\-y/d)/q z], ÏTIQ = 0, #o = 1 and (1) rrii = di-iqi-i - m^-i, (2) QiQi-i = d — m\ . (See for example in [1]). From these relations and some theorems concerning diophantine équations 0. Perron deduces in [2, p. 98] the famous resuit of Fermât: Every prime of the form 4m + 1 is a sum of two squares. In this note we will show that it is possible to do the same restricting theoretical tools to the above algorithm. Proof of the Two Square Theorem. The main idea is the same as Perron's and lies in the palindromatic nature of the fragments (mi , • • •, m s ) and (<7o, • • •, Çs), (see [2, p. 76]). In view of this and (2) d is a sum of two squares whenever s is odd. So, we'll be done showing that this is the case for the primes p = 4m + 1. Suppose then, that p = 1 (mod 4) and the length of the shortest period of the continued fraction expansion of yjp is even, say 5 = 2k. Then we have = m^+i and after some substitutions; (3) and (4) 2 m k = a kq k q k(±q k-\ + aUk) = 4p.
