Can You Prove that if n is an Integer > 1 then n^4 + 4 is never a prime? And what about Sophie Germain?
Tuesday 13 October 2026 at 21:01
Visible to anyone in the world
Edited by Richard Walker, Tuesday 13 October 2026 at 22:19
I found this in another book by Yakov Perelman, Algebra Can Be Fun (1979).
Here's a near solution
Write as . Now notice the first three terms are equal to , so we have .
But we know that for any and we have , the so-called 'Difference of two squares', and applying this to the expression above we get after rearranging slightly, . Providing these are both and thus we have found factors of and it cannot be prime.
Rather neat.
This identity is usually attributed to Sophie Germain, a mathematician and scientist, who amongst other things did important work on Fermat's Last Theorem in the early 19c.
When the Eiffel Tower was built the names of 72 (exclusively male) scientists were engraved on it. Now as a result of student campaigning and support from the Sorbonne, that wrong (as it seem to me to be) is being righted and the names of 72 female scientists including Sophie Germain have been approved and will be added to the monument in January 2027.
Can You Prove that if n is an Integer > 1 then n^4 + 4 is never a prime? And what about Sophie Germain?
I found this in another book by Yakov Perelman, Algebra Can Be Fun (1979).
Here's a near solution
Write as . Now notice the first three terms are equal to , so we have .
But we know that for any and we have , the so-called 'Difference of two squares', and applying this to the expression above we get after rearranging slightly, . Providing these are both and thus we have found factors of and it cannot be prime.