Richard Blecksmith, Michael McCallum, J. L. Selfridge,
... The University of Arizona, under the direction of John Brillhart. With his wife Sharon, daughter Katie, and various ...
Tópico(s): Mathematical and Theoretical Analysis
1998 - Taylor & Francis | American Mathematical Monthly
John Brillhart, Patrick Morton,
Tópico(s): Computability, Logic, AI Algorithms
1996 - Taylor & Francis | American Mathematical Monthly
Richard Blecksmith, John Brillhart, Irving Gerst,
We prove a six-parameter identity whose terms have the form x α T ( k 1 , l 1 ) T ( k 2 , l 2 ) {x^\alpha }T({k_1},{l_1})T({k_2},{l_2}) , where T ( k , l ) = ∑ − ∞ ∞ x k n 2 + l n T(k,l) = \sum \nolimits _{ - \infty }^\infty {{x^{k{n^2} + l\,n}}} . This identity is then used to give a new proof of the familiar Ramanujan identity H ( x ) G ( x 11 ) − x 2 G ( x ) H ( x 11 ) = 1 H(x)G({x^{11}}) - {x^2}G(x)H({x^{11}}) = 1 , where G ( x ) = ∏ n = 0 ∞ [ ( 1 − x 5 n + 1 ) ( 1 − x 5 n + 4 ) ] − 1 G(x) = \prod \nolimits _{n = 0}^\infty {{{[(1 - {x^{5n + 1}})(1 - {x^{5n + 4}})]}^{ - 1}}} and H ( x ) = ∏ n = 0 ∞ [ ( 1 − x 5 n + 2 ) ( 1 − x 5 n + 3 ) ] − 1 H(x) = \prod \nolimits _{ ...
Tópico(s): Finite Group Theory Research
1993 - American Mathematical Society | Mathematics of Computation
John Brillhart, D. H. Lehmer, J. L. Selfridge, Bryant Tuckerman, Samuel S. Wagstaff,
Tópico(s): Finite Group Theory Research
1988 - American Mathematical Society | Contemporary mathematics - American Mathematical Society
John Brillhart, Peter L. Montgomery, Robert Silverman,
We list the known prime factors of the Fibonacci numbers F n {F_n} for n ≤ 999 n \leq 999 and Lucas numbers L n {L_n} for n ≤ 500 n \leq 500 . We discuss the various methods used to obtain these factorizations, and primality tests, and give some history of the subject.
Tópico(s): Cryptographic Implementations and Security
1988 - American Mathematical Society | Mathematics of Computation
Richard Blecksmith, John Brillhart, Irving Gerst,
In this paper we derive the power series expansions of four infinite products of the form \[ ∏ n ∈ S 1 ( 1 − x n ) ∏ n ∈ S 2 ( 1 + x n ) , \prod \limits _{n \in {S_1}} {(1 - {x^n})\;\prod \limits _{n \in {S_2}} {(1 + {x^n}),} } \] where the index sets S 1 {S_1} and S 2 {S_2} are specified with respect to a modulus (Theorems 1, 3, and 4). We also establish a useful formula for expanding the product of two Jacobi triple products (Theorem 2). Finally, we give nonexistence results for identities of two forms.
Tópico(s): semigroups and automata theory
1988 - American Mathematical Society | Mathematics of Computation
Richard Blecksmith, John Brillhart, Irving Gerst,
In this paper we give a collection of parity results for partition functions of the form \[ ∏ n ∈ S ( 1 − x n ) − 1 ≡ ∑ − ∞ ∞ x e ( n ) ( mod 2 ) \prod _{n \in S} (1 - x^n)^{-1} \equiv \sum _{-\infty }^\infty x^{e(n)} \pmod 2 \] and \[ ∏ n ∈ S ( 1 − x n ) − 1 ≡ ∑ − ∞ ∞ ( x e ( n ) + x f ( n ) ) ( mod 2 ) \prod _{n \in S} (1 - x^n)^{-1} \equiv \sum _{-\infty }^\infty (x^{e(n)} + x^{f(n)})\pmod 2 \] for various sets of positive integers S , which are specified with respect to a modulus, and quadratic polynomials e ( n ) e(n) and f ( n ) f(n) . Several identities similar to theta function identities, such as \[ ∏ n = 1 n ≢ ± ( 4 , 6 , 8 , 10 ) ( ...
Tópico(s): Analytic Number Theory Research
1987 - American Mathematical Society | Mathematics of Computation
John Brillhart, P. Erdős, Richard Morton,
Let {a(n)} be the Rudin-Shapiro sequence, and let s(n) = Σ£ = o a ( k ) and t(n) = I" k=0 (-\) k a(k).In this paper we show that the sequences {s{n)/ Jn) and {t{n)/ Jn) do not have cumulative distribution functions, but do have logarithmic distribution functions (given by a specific Lebesgue integral) at each point of the respective intervals [γ/3/5, yfβ] and [0, V^] The functions a(x) and s(x) sore also defined for real x > 0, and the function [s(x) -a(x)]/ }/x is shown to have a Fourier expansion whose coefficients are ...
Tópico(s): Analytic Number Theory Research
1983 - Mathematical Sciences Publishers | Pacific Journal of Mathematics
... theorem was generalized in two different ways by John Brillhart, Andrew Odlyzko, and myself [ 1 ]. One way was ...
Tópico(s): Polynomial and algebraic computation
1982 - Cambridge University Press | Canadian Journal of Mathematics
The colon should be placed directly after the
Tópico(s): Matrix Theory and Algorithms
1982 - American Mathematical Society | Mathematics of Computation
John Brillhart, Michael Filaseta, Andrew Odlyzko,
In [ 1 , b.2, VIII, 128] Pólya and Szegö give the following interesting result of A. Cohn: THEOREM 1. If a prime p is expressed in the decimal system as then the polynomial irreducible in Z [ x ]. The proof of this result rests on the following theorem of Pólya and Szegö [ 1 , b.2, VIII, 127] which essentially states that a polynomial f(x) is irreducible if it takes on a prime value at an integer which is sufficiently far from the zeros of f(x) . THEOREM 2. Let f(x) ∊ Z [ x ] be a polynomial with the zeros α 1 , α 2 , …, α ...
Tópico(s): Meromorphic and Entire Functions
1981 - Cambridge University Press | Canadian Journal of Mathematics
Tópico(s): Electromagnetic Scattering and Analysis
1980 - Mathematical Sciences Publishers | Pacific Journal of Mathematics
John Brillhart, Patrick Morton,
The Rudin-Shapiro coefficients $\{a(n)\}$ are an infinite sequence of $\pm 1$'s, defined recursively by $a(0)=1$, $a(2n)=a(n)$, and $a(2n + 1)=(-1)^{n}a(n)$, $n \geq 0$ Various formulas are developed for the $n$th partial sum $s(n)$ and the $n$th alternating partial sum $t(n)$ of this sequence. These formulas are then used to show that $\sqrt{3/5} < s(n)/\surd n < \surd 6$ and $0 \leq; t(n)/\surd n < \surd 3$, $n \geq 1$ where the inequalities are sharp and the ratios are dense in the two intervals. For a given $n \geq 1$, the ...
Tópico(s): Graph theory and applications
1978 - Duke University Press | Illinois Journal of Mathematics
John Brillhart, J. S. Lomont, Patrick Morton,
Tópico(s): Mathematical functions and polynomials
1976 - De Gruyter | Journal für die reine und angewandte Mathematik (Crelles Journal)
John Brillhart, D. H. Lehmer, J. L. Selfridge,
Tópico(s): Mathematics and Applications
1975 - American Mathematical Society | Mathematics of Computation
Michael A. Morrison, John Brillhart,
The continued fraction method for factoring integers, which was introduced by D. H. Lehmer and R. E. Powers, is discussed along with its computer implementation. The power of the method is demonstrated by the factorization of the seventh Fermat number F 7 {F_7} and other large numbers of interest.
Tópico(s): Analytic Number Theory Research
1975 - American Mathematical Society | Mathematics of Computation
Michael A. Morrison, John Brillhart,
Tópico(s): Mathematics and Applications
1975 - American Mathematical Society | Mathematics of Computation
John Brillhart, D. H. Lehmer, J. L. Selfridge,
A collection of theorems is developed for testing a given integer N for primality. The first type of theorem considered is based on the converse of Fermat’s theorem and uses factors of N − 1 N - 1 . The second type is based on divisibility properties of Lucas sequences and uses factors of N + 1 N + 1 . The third type uses factors of both N − 1 N - 1 and N + 1 N + 1 and provides a more effective, yet more complicated, primality test. The search bound for factors of N ± 1 N \pm 1 and properties of the hyperbola N = x 2 − y ...
Tópico(s): Advanced Mathematical Theories and Applications
1975 - American Mathematical Society | Mathematics of Computation
Tópico(s): Advanced Mathematical Identities
1973 - Duke University Press | Duke Mathematical Journal
An improvement is given to the method of Hermite for finding a a and b b in p = a 2 + b 2 p = {a^2} + {b^2} , where p p is a prime ≡ 1 ( mod 4 ) {\text {prime}} \equiv 1\pmod 4 .
Tópico(s): Coding theory and cryptography
1972 - American Mathematical Society | Mathematics of Computation
(1971). On the Prime Divisors of Polynomials. The American Mathematical Monthly: Vol. 78, No. 3, pp. 250-266.
Tópico(s): Advanced Mathematical Theories
1971 - Taylor & Francis | American Mathematical Monthly
Tópico(s): Advanced Mathematical Identities
1971 - Taylor & Francis | American Mathematical Monthly
Michael A. Morrison, John Brillhart,
1971 - American Mathematical Society | Bulletin of the American Mathematical Society
where n > 0 and Po(x) = Qo(x) = 1. (See [4] also. Note in this reference that Po(x) =Qo(x) =x.) These polynomials have been used by Kahane and Salem in their book [1 ] to prove several theorems about trigonometric series. Rider [2] used a generalization of these polynomials to complete the solution of a problem partially solved in [4]. In a more recent paper Rider [3] employed the polynomials to exhibit certain subalgebras of the group algebra of the unit circle. In particular, in this paper Rider obtained a special ...
Tópico(s): Advanced Mathematical Identities
1970 - American Mathematical Society | Proceedings of the American Mathematical Society
Tópico(s): Advanced Mathematical Identities
1969 - De Gruyter | Journal für die reine und angewandte Mathematik (Crelles Journal)
Tópico(s): Polynomial and algebraic computation
1969 - Academic Press | Information and Control
The present paper is a completion of a previous paper of the same title (Zierler and Brillhart, 1968). In our preceding work 187 of the irreducible trinomials T~.k(x) = ~ x k ~- 1 were left to be tested for primitivity at a later date, even though the requisite complete factorizations of 2 ~ - 1 were known (these trinomials were identified in (Zierler and Brillhart, 1968) by a superscript minus sign on n). This testing has now been done on the CDC 6600 at the Communications Research Division of the Institute ...
Tópico(s): Polynomial and algebraic computation
1968 - Academic Press | Information and Control
John Brillhart, J. L. Selfridge,
Tópico(s): Varied Academic Research Topics
1967 - American Mathematical Society | Mathematics of Computation
This paper is the second of two papers dealing with the prime factors of Mersenne numbers Mp = 2P -1 of prime exponent (see Brillhart andJohnson [1]).The 899 new factors given below, which were discovered on the IBM 7090 of the Computing Facility at the University of California at Los Angeles, constitute the remaining factors needed for a complete listing in the literature of all prime factors q < 235 for 103 g p g 257 and q < 2U for 257 < p < 20000 (See Brillhart [2], Karst [4], Kravitz [5], Riesel [6], [7]).
Tópico(s): History and Theory of Mathematics
1964 - American Mathematical Society | Mathematics of Computation
THE CONGRUENCE d"-1 = 1 (mod p2) 149 John Brillhart.At the time of completion of these results ...
Tópico(s): Advanced Algebra and Geometry
1964 - American Mathematical Society | Mathematics of Computation