November 21, 2025International Journal of Number Theory0 citations

On sums of powers of consecutive squares over finite fields, and sums of distinct values of polynomials

Key Points

Determining sums of powers of squares reveals important relationships in finite fields.
The findings confirm conjectures related to sums of elements in image sets from polynomials.
Assessment involving elements of finite fields demonstrates new insights into the nature of powers and squares.
The results offer a fresh perspective on polynomial behaviors and their distinct values within finite fields.

Abstract

For each odd prime power Formula: see text, and each integer Formula: see text, we determine the sum of the Formula: see text-th powers of all elements Formula: see text for which both Formula: see text and Formula: see text are squares in Formula: see text. We also solve the analogous problem when one or both of Formula: see text and Formula: see text is a nonsquare. We use these results to determine the sum of the elements of the image set Formula: see text for each Formula: see text of the form Formula: see text, which resolves two conjectures by Finch-Smith, Harrington, and Wong.

Ask AI

Helpful

Bookmark

View Full Paper