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 r

2 + (a-+l)

2 53 Step 1: Determine all numbers N(x) for x = 1, 2,...,[(-1 + y/2M - l)/2]

. Step 2: Determine all R

n and R

n, as in Theorem 4.12 obtained from the Diophantine equations X

2 - 2

y

2 = 2

k

2 - 1, where k = 0,1,..., VF Step 3: Delete from the tabie of the numbers in Step 1, all numbers of Step 2. The remaining numbers are the only prime numbers of the form N(x) in the interval [5,M]. Proof. By using Theorem 4.12. References [1] C. F. GAUSS, Disqusitiones Arithmeticae, (English transl. A. A. Clarke, Yale University Press, New Haven, Connecticut, London, 1966.) [2] T. NAGELL, Introduction to Number Theory, Chelsea, New York, 1964. [3] W. SIERPINSKI, Sur les nombres triangulares qui sont sommes de deux nombres triangularies, Elem. Math., 17 (1962), 63-65. [4] P. G. TSANGARIS, Prime Numbers and Cyclotomy-Primes of the form z

2 + (x-fl)

2

, Ph.D. Thesis, Athens University, Athens, 1984 (in Greek). [5] P. G. TSANGARIS, Fermat-Pell Equation and the Numbers of the form ^

2+(iy+L)

Z

, Publ. Math. Debrecen, 47 (1995), 127-138. PANAYIOTIS G. TSANGARIS DEPARTMENT OF MATHEMATICS, ATHENS UNIVERSITY PANEPISTIMIOPOLIS, 157 84 ATHENS, GREECE.