Az Eszterházy Károly Tanárképző Főiskola Tudományos Közleményei. 1998. Sectio Mathematicae. (Acta Academiae Paedagogicae Agriensis : Nova series ; Tom. 25)

TSANGARIS, PANAYIOTIS G., A sieve for all primes of the form R2+(X-)-l)2 ....

A sieve for all primes of the form x

2 + (.r+i)

2 PANAYIOTIS G. TSANGARIS Abstract: All composite numbers of the form x

2+(x+l)

2 are determined in terms of suitable (non-homogeneous) linear recurrence sequences of order 2 (Theorem 4.12). As a consequence, all primes of the same form in a given interval can be determined by a sieving procedure (Theorem 4.13). Introduction The object of this study axe the prime and composite numbers of the form x

2 -f (x + l)

2. Their study depends heavily on the following Theorem 1.1. (SlERPiNSKl) [3]) The number x

2 + (x -f l)

2 is composite if and only if there exist natural numbers y

, 2 such that: (T

) T(x) = T(y) + T(z). (Here T(x),T(y)

1T(z) denote triangular numbers

.) The description of all composite numbers of the form x

2 + (x -f I)

2 is reduced to the study of the integral solutions of the following family of Diophantine equations of Ferinat-Pell type: (F

k) X

2 - 2Y

2 = 2k

2 - 1, k = 0,1,2,.... Thus the study of equation (T) is reduced to the study of the family of equations (Fk) in terms of Gauss type transformations. The detailed study of all solutions of (Fk) is carried on via Nagell's method of equivalence classes, thus avoiding any reference to fundamental units. We will consider the Diophantine equation (1.1) (

2-dr,

2 = -l (df •) where d / • (non-square) is a natural number. The sequence of non-negative (that is £271+1 ^ 0 and 77271+1 — hitegral solutions of (1.1) is determined by the following recursive formulae: £271+3 = 2x\ £271+1 - £271—1 ? where ft = £1 and £

3 = £? + 3d£i r}\ \ / 2 3 7/2n+3 = 2x

l7]

2n

+i - T]

2

n-1, where r)i = t]i and r/

3 = 771 + dr)

x,