Arithmetic progressions in the values of a quadratic polynomial

Abstract

The following theorem is proved: THEOREM.Let A, B, C, A ¥= 0 be rational numbers.There do not exist four unequal rational numbers x 1? x 2 , x 3 , x 4 such that /(Xj), /( X 2)> /( X 3)> f( x 4) are * n arithmetic progression, whereThe proof depends on determining the rational points on a certain elliptic curve.This paper is concerned with the proof of the following theorem.THEOREM.Let A, B, C, A ¥= 0 be rational numbers.There do not exist four unequal rational numbers x lt x 2 , x 3 , x 4 such that /(a^), f(x 2 ), f(x 3 ), f(x 4 ) are in arithmetic progression, where f(x) = Ax 2 + Bx -f C. PROOF.Assuming the contrary, we may normalize the x i and / so that Xi = 0, 1, a, b while /(O) = 0, /(I) = 1, /(a) = 2, f(b) = 3.For a quadratic polynomial to satisfy these relations, it is necessary that

Locations

• Rocky Mountain Journal of Mathematics - View - PDF