ultimate generalisation of Euler-Fermat theorem

Let $a^{b}+u=m$ where $a,\,b,\,u,\,m$ are positive integers. Then

 $a^{b+k\varphi(m)}+u\;\equiv\;0\pmod{m},$

by the result in “Euler’s generalisation of Fermat’s theorem – a further generalisation”. Proceedings of Hawaii Intl. conference on maths & statistics 2004 (ISSN 1550–3747). Here, $k$ is a positive integer. Next,

 $a^{b^{1+k\varphi(\varphi(m))}}+u\;\equiv\;0\pmod{m}.$

(This is a corollary of “Euler’s generalisation of Fermat’s theorem – a further generalisation”.) We can proceed in a like manner till we reach

 $a^{b^{c^{\vdots^{t^{1+k\varphi(\varphi(\varphi(\ldots\varphi(2)\ldots)))}}}}}.$

At this stage onwards the function generates only multiples of $m$ and no prime number is generated. This is the ultimate generalisation of Fermat’s theorem. Please note that each step of multiple exponentiation in the above is a corollary of the theorem referred to.

Title ultimate generalisation of Euler-Fermat theorem UltimateGeneralisationOfEulerFermatTheorem 2013-03-22 19:35:04 2013-03-22 19:35:04 akdevaraj (13230) akdevaraj (13230) 5 akdevaraj (13230) msc 11A99