Let W(n):=an^2+bn+c where a,b,c in ZZ.

Jaslyn Sloan

Jaslyn Sloan

Answered question

2022-11-13

Let W ( n ) := a n 2 + b n + c where a , b , c Z
Assume that for all n Z we have that W(n) is the square of an integer.
Show that there exists some P such that W ( n ) = P ( n ) 2

Answer & Explanation

lavarcar2d2

lavarcar2d2

Beginner2022-11-14Added 18 answers

Without loss of generality, you can assume a , b , c 0. Since
4 a W ( x ) ( 2 a x + b ) 2 = 4 a c b 2 = D ,
we have that the Pell equation
4 a u 2 v 2 = D ,
for any N big enough, has at least N 2 a integer solutions (u,v) with | v | N. From the theory of Pell equation we know that, if D 0, there are at most O ( log N ) solutions with | v | N, so
D = 0 , a = A 2 , c = W ( 0 ) = C 2 , W ( x ) = ( A x + C ) 2
must hold.

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