Ahmet Tekcan

On the number of representations of positive integers by quadratic forms as the basis of the space S4(Γ0(47),1)

The number of representations of positive integers by quadratic forms F 1 = x 1 2 + x 1 x 2 + 12 x 2 2 and G 1 = 3 x 1 2 + x 1 x 2 + 4 x 2 2 of discriminant − 47 are given. Moreover, a basis for the space S 4 ( Γ 0 ( 47 ) , 1 ) are constructed, and the formulas for r ( n ; F 4 ) , r ( n ; G 4 ) , r ( n ; F 3 ⊕ G 1 ) , r ( n ; F 2 ⊕ G 2 ) , and r ( n ; F 1 ⊕ G 3 ) are derived.

The Diophantine Equation and -Balancing Numbers

Let be an odd integer such that is a prime. In this work, we determine all integer solutions of the Diophantine equation and then we deduce the general terms of all -balancing numbers.

Indefinite Quadratic Forms and their Neighbours

The aim of this work is to derive the connection among cycle, proper cycle, right and left neighbour of indefinite reduced binary quadratic form F(x,y) = ax2 +bxy+cy2 of discriminant Δ = b2−4ac.

BASE POINTS, BASES AND POSITIVE DEFINITE FORMS

In this work, we deduce some algebraic relations on base points z(F) of positive definite forms F = (a,b,c) and ℝ-basis of ℂ.

Integer Solutions of a Special Diophantine Equation

Let t≠1 be an integer. In this work, we determine the integer solutions of Diophantine equation D:x2+(2−t2)y2+(−2t2−2t+2)x+(2t5−6t3+4t)y−t8+4t6−4t4+2t3+t2−2t = 0 over Z and also over finite fields Fp for primes p≥2. Also we derive some recurrence relations on the integer solutions (xn,yn) of D and formulate the the n—th solution (xn,yn) by using the simple continued fraction expansion of xnyn.

Primes in Z[exp(2πi/3)]

In this paper, we study the primes in the ring Z[w], where w = exp(2πi/3) is a cubic root of unity. We gave a classification of them and some results related to the use of them in the calculation of cubic residues are obtained.