Az Eszterházy Károly Tanárképző Főiskola Tudományos Közleményei. 1994. Sectio Mathematicae. (Acta Academiae Paedagogicae Agriensis : Nova series ; Tom. 22)

HERENDI, T. and PETHŐ, A., Trinomials, which are divisible by quadratic polynomials

Trinomials, which are divisible by quadratic polynomials 65 3. Basic lemmata The following lemma generalizes a result of Ribenboim [4] and it is basic for the proofs of the theorems. Lemma 4. Let n > 2, 1 < k < n and a, 6, A, B £ Z. If x

2 — bx — a divides x

n - Bx

k - A then B • b

k~

l mod 2 • f

k.

x (a, b

2) = b

n~

l mod 2 • fn—i (a, b

2) . Further if (a) b

k~

l mod

2 • /

f

c_! (a, b

2) = 0 then b

n~

l mod

2 • /

n_

x (a,6

2) = 0 and A-a - (b

n~

2 mod 2 • f

n2 (a,b

2)-B• ™

d 2 • (a, ö

2)) . (b) otherwise 6"1 mod 2 ' fn—i (a, 6

2) fl

| ' mod

2 • A_i (a,6

2) and A = a

f

c( —1) fc, ^

kb

nhlmod

2-fn-k-i{a,b

2) bk-1 mod 2 * PROOF. (a) Assume that a;

7

1 — Bx

k — A — (a;

2 — bx — a) • p(x) with p(x) = xn-2 Cn 3x

n~

3 -f c

n_

4:r

n~

4 + • • • + c\x -f Co- Similarly as in [4] we have the following équations: A — a • CQ • B = a • ci + b • c

0 (2) • 5 = a • Ci + b • Ci-1 - Ci_

2 ^n-2,Jb • 5 = a + 6 • c

n_

3 - c

n_4 ^n-l,* B = b - C

n_3 where = { _ / 1 if i = j 0 otherwise