By Wieb Bosma, John Cannon

This quantity celebrates the 1st decade of the pc Algebra approach Magma. With a layout in accordance with the ontology and semantics of algebra, Magma allows clients to swiftly formulate and practice calculations within the extra summary elements of arithmetic. This ebook introduces the reader to the position Magma performs in complicated mathematical examine via 14 case stories which, more often than not, describe computations underpinning new theoretical effects. The authors of the chapters have been selected either for his or her services within the specific box and for his or her cutting edge use of Magma. even though in no way exhaustive, the subjects diversity over a lot of Magma's insurance of algorithmic algebra: from quantity thought and algebraic geometry, through illustration conception and workforce concept to a few branches of discrete arithmetic and graph idea. A simple creation to the Magma language is given in an appendix. The ebook is concurrently a call for participation to profit a brand new programming language within the context of latest study difficulties, and an exposition of the categories of challenge that may be investigated utilizing computational algebra.

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Richard K. Guy, Unsolved problems in number theory, Unsolved problems in intuitive mathematics I, New York: Springer 1994 (2nd edition). 21. Kenneth Ireland and Michael Rosen, A classical introduction to modern number theory, Graduate texts in mathematics 84, New York: Springer, 1982. 22. G. Jaeschke, On the smallest k such that all k · 2N + 1 are composite, Math. Comp. 40 (1983), 381–384; Errata: Math. Comp. 45 (1985), 637. 23. Wilfrid Keller, Factors of Fermat numbers and large primes of the form k ·2n +1, Math.

The smallest value k for which no m was encountered with #φ−1 (m) = k was k = 4077. It is an open conjecture that every k > 1 will occur eventually. Erd˝ os proved [17] that if there exists an integer m for which #φ−1 (m) = k, then there exist inﬁnitely many such m. This was done by a fairly complicated analytic argument, showing that there are very many primes p such that #φ−1 ((p − 1)m) = #φ−1 (m) = k. Iteration of φ ◦ σ Another conjecture about φ concerns the iteration of the composite φ ◦ σ of φ and the divisor-σ function, which assigns to n the sum of its divisors σ(n) = d|n d.

The presentation for function ﬁelds focuses mainly on the diﬀerences between number ﬁelds and function ﬁelds as they essentially behave in the same way. We introduce divisors, divisor class groups and divisor ray class groups. As applications of the class ﬁeld theory developed, we show how to compile certain tables of global ﬁelds indexed by degree, discriminant and Galois group. For function ﬁelds, we show how the theory may be used to ﬁnd ﬁelds with “many rational places” which are of interest in certain applications such as in the construction of “good codes”.