Department of Mathematics Publications

Euler’s Prime-Producing Polynomial Revisited

Robert Heffernan, Department of Mathematics, Munster Technological University, Cork, IrelandFollow
Nick Lord, Tonbridge School, Kent TN9 1JP
Des MacHale, School of Mathematics, Applied Mathematics and Statistics, University College Cork, Cork, Ireland

Document Type

Article

Creative Commons License

This work is licensed under a Creative Commons Attribution-NonCommercial-Share Alike 4.0 International License.

Disciplines

Physical Sciences and Mathematics

Publication Details

The Mathematical Gazette, vol 108, no. 571. © The Authors, 2024

Abstract

Euler’s polynomial f (n) = n2 + n + 41 is famous for producing 40 different prime numbers when the consecutive values 0, 1, …, 39 are substituted: see Table 1. Some authors, including Euler, prefer the polynomial f (n − 1) = n2 − n + 41 with prime values for n = 1, …, 40. Since f (−n) = f (n − 1), f (n) actually takes prime values (with each value repeated once) for n = −40, −39, …, 39; equivalently the polynomial f (n − 40) = n2 − 79n + 1601 takes (repeated) prime values for n = 0, 1, …, 79.