On the Analytical Properties of Prime Numbers
In this work we have studied the prime numbers in the model P=am+1,m,a>1∈N. and the number in the form q=mam+bm+1in particular, we provided tests for hem. This is considered a generalization of the work José María Grau and Antonio M. Oller-marcén prove that if Cma=mam+1 is a generalized Cullen nu...
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| Formato: | Capítulo de libro |
| Lenguaje: | inglés |
| Publicado: |
IntechOpen
2023
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| Materias: | |
| Acceso en línea: | https://www.intechopen.com/chapters/87471 |
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| Sumario: | In this work we have studied the prime numbers in the model P=am+1,m,a>1∈N. and the number in the form q=mam+bm+1in particular, we provided tests for hem. This is considered a generalization of the work José María Grau and Antonio M. Oller-marcén prove that if Cma=mam+1 is a generalized Cullen number then mam≡−1amodCma. In a second paper published in 2014, they also presented a test for Broth’s numbers in Form kpn+1 where k<pn. These results are basically a generalization of the work of W. Bosma and H.C Williams who studied the cases, especially when p=2,3, as well as a generalization of the primitive MillerRabin test. In this study in particular, we presented a test for numbers in the form mam+bm+1in the form of a polynomial that highlights the properties of these numbers as well as a test for the Fermat and Mersinner numbers and p=ab+1a,b>1∈Nand p=qa+1where qis primeoddare special cases of the number mam+bm+1when btakes a specific value. For example, we proved if p=qa+1where q is odd prime and a>1∈N where πj=1qqjthen ∑j=1q−2πj−Cmaq−j−1q−am≡χmq−ammodp Components of proof Binomial theorem Fermat’s Litter Theorem Elementary algebra. |
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| ISBN: | 978-1-80356-953-6 |
| DOI: | 10.5772/intechopen.109365 |