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单词 UltimateGeneralisationOfEulerFermatTheorem
释义

ultimate generalisation of Euler-Fermat theorem


Let ab+u=m where a,b,u,m are positive integers. Then

ab+kφ(m)+u 0(modm),

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,

ab1+kφ(φ(m))+u 0(modm).

(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

abct1+kφ(φ(φ(φ(2)))).

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

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更新时间:2025/5/4 3:42:05