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Euler’s prime-producing polynomial revisited

Published online by Cambridge University Press:  15 February 2024

Robert Heffernan ,
Nick Lord  and
Des MacHale
Robert Heffernan
Affiliation:
Department of Mathematics, Munster Technological University, Cork, Ireland e-mail: Robert.Heffernan@mtu.ie
Nick Lord
Affiliation:
Tonbridge School, Kent TN9 1JP e-mail: njl@tonbridge-school.org
Des MacHale
Affiliation:
School of Mathematics, Applied Mathematics and Statistics, University College Cork, Cork, Ireland e-mail: d.machale@ucc.ief
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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.

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Articles
Information
The Mathematical Gazette , Volume 108 , Issue 571 , March 2024 , pp. 69 - 77
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This is an Open Access article, distributed under the terms of the Creative Commons Attribution-NonCommercial-ShareAlike licence (https://creativecommons.org/licenses/by-nc-sa/4.0/), which permits non-commercial re-use, distribution, and reproduction in any medium, provided the same Creative Commons licence is included and the original work is properly cited. The written permission of Cambridge University Press must be obtained for commercial re-use.
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© The Authors, 2024 Published by Cambridge University Press on behalf of The Mathematical Association

References

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