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.
Tags: Sophie Germain, difference of two squares, Eiffel Tower, Yakov Perelman
