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Sister Beiter Conjecture
In mathematics, the Sister Beiter conjecture is a conjecture about the size of coefficients of ternary cyclotomic polynomials (i.e. where the index is the product of three prime numbers). It is named after Marion Beiter Sister Marion Beiter (August 23, 1907 – October 11, 1982), born Dorothy Katharine Beiter, was an American mathematician and educator. Her research focused on the area of cyclotomic polynomials. Beiter was born in Buffalo to Kathryn () and ..., a Catholic nun who first proposed it in 1968. Background For n\in\mathbb_ the maximal coefficient (in absolute value) of the cyclotomic polynomial \Phi_n(x) is denoted by A(n). Let 3\leq p\leq q\leq r be three prime numbers. In this case the cyclotomic polynomial \Phi_(x) is called ''ternary''. In 1895, A. S. Bang proved that A(pqr)\leq p-1. This implies the existence of M(p):=\max\limits_A(pqr) such that 1\leq M(p)\leq p-1. Statement Sister Beiter conjectured in 1968 that M(p)\leq \frac. This was later disproved, ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Cyclotomic Polynomial
In mathematics, the ''n''th cyclotomic polynomial, for any positive integer ''n'', is the unique irreducible polynomial with integer coefficients that is a divisor of x^n-1 and is not a divisor of x^k-1 for any Its roots are all ''n''th primitive roots of unity e^ , where ''k'' runs over the positive integers not greater than ''n'' and coprime to ''n'' (and ''i'' is the imaginary unit). In other words, the ''n''th cyclotomic polynomial is equal to : \Phi_n(x) = \prod_\stackrel \left(x-e^\right). It may also be defined as the monic polynomial with integer coefficients that is the minimal polynomial over the field of the rational numbers of any primitive ''n''th-root of unity ( e^ is an example of such a root). An important relation linking cyclotomic polynomials and primitive roots of unity is :\prod_\Phi_d(x) = x^n - 1, showing that is a root of x^n - 1 if and only if it is a ''d''th primitive root of unity for some ''d'' that divides ''n''. Examples If ''n'' is a p ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Marion Beiter
Sister Marion Beiter (August 23, 1907 – October 11, 1982), born Dorothy Katharine Beiter, was an American mathematician and educator. Her research focused on the area of cyclotomic polynomials. Beiter was born in Buffalo to Kathryn () and Edward Frederick Beiter, where she attended Sacred Heart Academy. She entered the Sisters of St. Francis of Penance and Christian Charity in 1923, and professed her final vows in 1929. She began her career in 1925 as a teacher in parochial and private schools, continuing in this capacity until 1952, when she was appointed chairwoman of the mathematics department of Rosary Hill College. She meanwhile graduated from Canisius College (1944) and St. Bonaventure University (1948), before obtaining a PhD from the Catholic University of America in 1960. In her work on cyclotomic polynomials and their coefficients she made a conjecture referred to as Sister Beiter conjecture. Besides a sabbatical year at the State University of New York at Bu ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
The American Mathematical Monthly
''The American Mathematical Monthly'' is a mathematical journal founded by Benjamin Finkel in 1894. It is published ten times each year by Taylor & Francis for the Mathematical Association of America. The ''American Mathematical Monthly'' is an expository journal intended for a wide audience of mathematicians, from undergraduate students to research professionals. Articles are chosen on the basis of their broad interest and reviewed and edited for quality of exposition as well as content. In this the ''American Mathematical Monthly'' fulfills a different role from that of typical mathematical research journals. The ''American Mathematical Monthly'' is the most widely read mathematics journal in the world according to records on JSTOR. Tables of contents with article abstracts from 1997–2010 are availablonline The MAA gives the Lester R. Ford Awards annually to "authors of articles of expository excellence" published in the ''American Mathematical Monthly''. Editors *2022� ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Conjectures About Prime Numbers
In mathematics, a conjecture is a conclusion or a proposition that is proffered on a tentative basis without proof. Some conjectures, such as the Riemann hypothesis (still a conjecture) or Fermat's Last Theorem (a conjecture until proven in 1995 by Andrew Wiles), have shaped much of mathematical history as new areas of mathematics are developed in order to prove them. Important examples Fermat's Last Theorem In number theory, Fermat's Last Theorem (sometimes called Fermat's conjecture, especially in older texts) states that no three positive integers a, ''b'', and ''c'' can satisfy the equation ''a^n + b^n = c^n'' for any integer value of ''n'' greater than two. This theorem was first conjectured by Pierre de Fermat in 1637 in the margin of a copy of ''Arithmetica'', where he claimed that he had a proof that was too large to fit in the margin. The first successful proof was released in 1994 by Andrew Wiles, and formally published in 1995, after 358 years of effort by mat ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
