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P-Square are a Nigerian music duo composed of the twin brothers Peter Okoye and Paul Okoye, who co-write and co-produce most of their songs. [4] [5] [6] Noted for their artistic reinvention, musical versatility, and visual presentation, they are widely regarded as one of the most influential African acts of all time and one of the most successful music groups from Africa.
The observations about −3 and 5 continue to hold: −7 is a residue modulo p if and only if p is a residue modulo 7, −11 is a residue modulo p if and only if p is a residue modulo 11, 13 is a residue (mod p) if and only if p is a residue modulo 13, etc. The more complicated-looking rules for the quadratic characters of 3 and −5, which ...
The last step uses the fact that p 2 divides 2 p(p−1) − 1. This follows from Fermat's little theorem, which shows that, for p > 2, 2 p−1 = pk + 1 for some integer k. Raising both sides to the power of p then shows that 2 p(p−1) = p 2 (...) + 1. And now with a similar calculation as above, the following results:
[2] The seven problems were officially announced by John Tate and Michael Atiyah during a ceremony held on May 24, 2000 (at the amphithéâtre Marguerite de Navarre) in the Collège de France in Paris. [3] Grigori Perelman, who had begun work on the Poincaré conjecture in the 1990s, released his proof in 2002 and 2003. His refusal of the Clay ...
Pseudomathematics, or mathematical crankery, is a mathematics-like activity that does not adhere to the framework of rigor of formal mathematical practice. Common areas of pseudomathematics are solutions of problems proved to be unsolvable or recognized as extremely hard by experts, as well as attempts to apply mathematics to non-quantifiable ...
The above definition states that a composite integer n is Carmichael precisely when the nth-power-raising function p n from the ring Z n of integers modulo n to itself is the identity function. The identity is the only Z n - algebra endomorphism on Z n so we can restate the definition as asking that p n be an algebra endomorphism of Z n .
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The group (/) is cyclic if and only if n is 1, 2, 4, p k or 2p k, where p is an odd prime and k > 0. For all other values of n the group is not cyclic. [ 1 ] [ 2 ] [ 3 ] This was first proved by Gauss .