Riele, H. J. J. te 1947
Overview
Works:  102 works in 275 publications in 3 languages and 850 library holdings 

Genres:  Conference papers and proceedings 
Roles:  Author, Editor, Other 
Classifications:  QA76.6, 005.1 
Publication Timeline
.
Most widely held works by
H. J. J. te Riele
Algorithms and applications on vector and parallel computers by
H. J. J. te Riele(
Book
)
9 editions published in 1987 in English and Italian and held by 199 WorldCat member libraries worldwide
9 editions published in 1987 in English and Italian and held by 199 WorldCat member libraries worldwide
A theoretical and computational study of generalized aliquot sequences by
H. J. J. te Riele(
Book
)
15 editions published between 1975 and 1976 in English and Undetermined and held by 141 WorldCat member libraries worldwide
15 editions published between 1975 and 1976 in English and Undetermined and held by 141 WorldCat member libraries worldwide
European Congress of Mathematics, Amsterdam, 1418 July, 2008 by
2008, Amsterdam> European Congress of Mathematics. <5(
Book
)
6 editions published in 2010 in English and held by 65 WorldCat member libraries worldwide
6 editions published in 2010 in English and held by 65 WorldCat member libraries worldwide
Colloquium Numerical Treatment of Integral Equations by Workshop on Numerical Treatment of Integral Equations(
Book
)
6 editions published in 1979 in English and held by 39 WorldCat member libraries worldwide
6 editions published in 1979 in English and held by 39 WorldCat member libraries worldwide
New experimental results concerning the Goldbach conjecture by
JeanMarc Deshouillers(
Book
)
5 editions published in 1998 in English and Spanish and held by 13 WorldCat member libraries worldwide
Abstract: "The Goldbach conjecture states that every even integer [> or =]4 can be written as a sum of two prime numbers. It is known to be true up to 4 X 10¹¹. In this paper, new experiments on a Cray C916 supercomputer and on an SGI compute server with 18 R8000 CPUs are described, which extend this bound to 10¹⁴. Two consequences are that (1) under the assumption of the Generalized Riemann hypothesis, every odd number [> or =]7 can be written as a sum of three prime numbers, and (2) under the assumption of the Riemann hypothesis, every even positive integer can be written as a sum of at most four prime numbers. In addition, we have verified the Goldbach conjecture for all the even numbers in the intervals [10[superscript 5i], 10[superscript 5i] + 10⁸], for i=3,4, ...,20 and [10[superscript 10i], 10[superscript 10i] + 10⁹], for i=20,21, ... 30. A heuristic model is given which predicts the average number of steps needed to verify the Goldbach conjecture on a given interval. Our experimental results are in good agreement with this prediction. This adds to the evidence of the truth of the Goldbach conjecture."
5 editions published in 1998 in English and Spanish and held by 13 WorldCat member libraries worldwide
Abstract: "The Goldbach conjecture states that every even integer [> or =]4 can be written as a sum of two prime numbers. It is known to be true up to 4 X 10¹¹. In this paper, new experiments on a Cray C916 supercomputer and on an SGI compute server with 18 R8000 CPUs are described, which extend this bound to 10¹⁴. Two consequences are that (1) under the assumption of the Generalized Riemann hypothesis, every odd number [> or =]7 can be written as a sum of three prime numbers, and (2) under the assumption of the Riemann hypothesis, every even positive integer can be written as a sum of at most four prime numbers. In addition, we have verified the Goldbach conjecture for all the even numbers in the intervals [10[superscript 5i], 10[superscript 5i] + 10⁸], for i=3,4, ...,20 and [10[superscript 10i], 10[superscript 10i] + 10⁹], for i=20,21, ... 30. A heuristic model is given which predicts the average number of steps needed to verify the Goldbach conjecture on a given interval. Our experimental results are in good agreement with this prediction. This adds to the evidence of the truth of the Goldbach conjecture."
Computational sieving applied to some classical numbertheoretic problems by
H. J. J. te Riele(
Book
)
5 editions published in 1998 in English and held by 12 WorldCat member libraries worldwide
Abstract: "Many problems in computational number theory require the application of some sieve. Efficient implementation of these sieves on modern computers has extended our knowledge of these problems considerably. This is illustrated by three classical problems: the Goldback conjecture, factoring large numbers, and computing the summatory function of the Möbius function."
5 editions published in 1998 in English and held by 12 WorldCat member libraries worldwide
Abstract: "Many problems in computational number theory require the application of some sieve. Efficient implementation of these sieves on modern computers has extended our knowledge of these problems considerably. This is illustrated by three classical problems: the Goldback conjecture, factoring large numbers, and computing the summatory function of the Möbius function."
A new method for finding amicable pairs by
H. J. J. te Riele(
Book
)
4 editions published in 1995 in English and held by 11 WorldCat member libraries worldwide
Abstract: "Let [sigma](x) denote the sum of all divisors of the (positive) integer x. An amicable pair is a pair of integers (m, n) with m <n such that [sigma](m) = [sigma](n) = m + n. The smallest amicable pair is (220,284). A new method for finding amicable pairs is presented, based on the following observation of Erdős: For given s, let x₁, x₂ ... be solutions of the equation [sigma](x) = s, then any pair (x[subscript i], x[subscript j]) for which x[subscript i] + x[subscript j] = s is amicable. The problem here is to find numbers s for which the equation [sigma](x) = s has many solutions. From inspection of tables of known amicable pairs and their pair sums one learns that certain smooth numbers s (i.e., numbers with only small prime divisors) are good candidates. With the help of a precomputed table of [sigma](p[superscript e])values, many solutions of the equation [sigma](x) = s were found by checking divisibility of s by the tabled [sigma]values in a recursive way. In the set of solutions found, pairs were traced which sum up to s. From 1850 smooth numbers s satisfying 4 x 10¹¹ <s <10¹² we found 116 new amicable pairs with this algorithm. After the submission of this paper to the Vancouver Conference Mathematics of Computation 19431993, the computations have been extended and yielded many more new amicable pairs. In particular, the first quadruple of amicable pairs with the same pair sum (namely 16!) was found. A list is given of 587 amicable pairs with smaller member [sic] between 2.01 x 10¹¹ and 10¹², of which 565 pairs seem to be new."
4 editions published in 1995 in English and held by 11 WorldCat member libraries worldwide
Abstract: "Let [sigma](x) denote the sum of all divisors of the (positive) integer x. An amicable pair is a pair of integers (m, n) with m <n such that [sigma](m) = [sigma](n) = m + n. The smallest amicable pair is (220,284). A new method for finding amicable pairs is presented, based on the following observation of Erdős: For given s, let x₁, x₂ ... be solutions of the equation [sigma](x) = s, then any pair (x[subscript i], x[subscript j]) for which x[subscript i] + x[subscript j] = s is amicable. The problem here is to find numbers s for which the equation [sigma](x) = s has many solutions. From inspection of tables of known amicable pairs and their pair sums one learns that certain smooth numbers s (i.e., numbers with only small prime divisors) are good candidates. With the help of a precomputed table of [sigma](p[superscript e])values, many solutions of the equation [sigma](x) = s were found by checking divisibility of s by the tabled [sigma]values in a recursive way. In the set of solutions found, pairs were traced which sum up to s. From 1850 smooth numbers s satisfying 4 x 10¹¹ <s <10¹² we found 116 new amicable pairs with this algorithm. After the submission of this paper to the Vancouver Conference Mathematics of Computation 19431993, the computations have been extended and yielded many more new amicable pairs. In particular, the first quadruple of amicable pairs with the same pair sum (namely 16!) was found. A list is given of 587 amicable pairs with smaller member [sic] between 2.01 x 10¹¹ and 10¹², of which 565 pairs seem to be new."
Computer verification of the AnkenyArtinChowla conjecture for all primes less than 100 000 000 000 by
A. J Van Der Poorten(
Book
)
4 editions published in 1999 in English and held by 11 WorldCat member libraries worldwide
Let p be a prime congruent to 1 modulo 4 and let t, u be rational integers such that (t+u por raiz de p)/2 is the fundamental unit of the real quadratic field Q (raiz de p). The AnkenyArtinChowla conjecture (AACC) asserts that p will not divide u. This is equivalent to the assertation that p will no divided B sub (p1)/2, where B sub n denotes the nth Bernoulli number. Although first published in 1952, this conjecture still remains unproved today. Indeed, it appears to be most difficult to prove. Even testing the conjecture can be quite challenging because of the size of the numbers t, u; for example, when p=40 094 470 441, then both t and u exceed 10 elevado a la 330000. In 1988 the AAC conjecture was verified by computer for all p10 elevado a la 9 . In this paper we describe a new technique for testing the AAC conjecture and we provide some results of a computer run of the method for all primes p up to 100 elevado a la 11
4 editions published in 1999 in English and held by 11 WorldCat member libraries worldwide
Let p be a prime congruent to 1 modulo 4 and let t, u be rational integers such that (t+u por raiz de p)/2 is the fundamental unit of the real quadratic field Q (raiz de p). The AnkenyArtinChowla conjecture (AACC) asserts that p will not divide u. This is equivalent to the assertation that p will no divided B sub (p1)/2, where B sub n denotes the nth Bernoulli number. Although first published in 1952, this conjecture still remains unproved today. Indeed, it appears to be most difficult to prove. Even testing the conjecture can be quite challenging because of the size of the numbers t, u; for example, when p=40 094 470 441, then both t and u exceed 10 elevado a la 330000. In 1988 the AAC conjecture was verified by computer for all p10 elevado a la 9 . In this paper we describe a new technique for testing the AAC conjecture and we provide some results of a computer run of the method for all primes p up to 100 elevado a la 11
Some experiences of solving 1D semiconductor device equations on a Matrix coprocessor by a domain decomposition method by
C. H Lai(
Book
)
4 editions published in 1993 in English and held by 10 WorldCat member libraries worldwide
Abstract: "In this report we implement a domain decomposition technique for the numerical solution of 1D semiconductor device equations on a Cray SMP System 500 Matrix Coprocessor with 28 processing elements. A total work expression is constructed for comparison with the actual computing time of the parallel technique. We examine the behaviour of the numerical method by using different configurations of the processing elements within the parallel machine. We perform experiments on a number of devices including pn junctions and thyristors."
4 editions published in 1993 in English and held by 10 WorldCat member libraries worldwide
Abstract: "In this report we implement a domain decomposition technique for the numerical solution of 1D semiconductor device equations on a Cray SMP System 500 Matrix Coprocessor with 28 processing elements. A total work expression is constructed for comparison with the actual computing time of the parallel technique. We examine the behaviour of the numerical method by using different configurations of the processing elements within the parallel machine. We perform experiments on a number of devices including pn junctions and thyristors."
Tables of the first 15000 zeros of the Riemann zeta function to 28 significant digits, and related quantities by
H. J. J. te Riele(
Book
)
4 editions published in 1979 in English and held by 10 WorldCat member libraries worldwide
4 editions published in 1979 in English and held by 10 WorldCat member libraries worldwide
Factoring integers with large prime variations of the quadratic sieve by
H Boender(
Book
)
3 editions published in 1995 in English and held by 9 WorldCat member libraries worldwide
Abstract: "We present the results of many factorization runs with the single and double large prime variations (PMPQS, and PPMPQS, respectively) of the quadratic sieve factorization method on SGI workstations, and on a Cray C90 vectorcomputer. Experiments with 71, 87, and 99digit numbers show that for our Cray C90 implementations PPMPQS beats PMPQS for numbers of more than 80 digits, and this crossover point goes down with the amount of available central memory. For PMPQS a known theoretical formula is worked out and tested that helps to predict the total running time on the basis of a short test run. The accuracy of the prediction is within 10% of the actual running time. For PPMPQS such a prediction formula is not known and the determination of an optimal choice of the parameters for a given number would require many full runs with that given number, and the use of an inadmissible amount of CPUtime. In order yet to provide measurements that can help to determine a good choice of the parameters in PPMPQS, we have factored many numbers in the 66  88 decimal digits range, where each number was run once with a specific choice of the parameters. In addition, an experimental prediction formula is given that has a restricted scope in the sense that it only applies to numbers of a given size, for a fixed choice of the parameters of PPMPQS. So such a formula may be useful if one wishes to factor many different large numbers of about the same size with PPMPQS."
3 editions published in 1995 in English and held by 9 WorldCat member libraries worldwide
Abstract: "We present the results of many factorization runs with the single and double large prime variations (PMPQS, and PPMPQS, respectively) of the quadratic sieve factorization method on SGI workstations, and on a Cray C90 vectorcomputer. Experiments with 71, 87, and 99digit numbers show that for our Cray C90 implementations PPMPQS beats PMPQS for numbers of more than 80 digits, and this crossover point goes down with the amount of available central memory. For PMPQS a known theoretical formula is worked out and tested that helps to predict the total running time on the basis of a short test run. The accuracy of the prediction is within 10% of the actual running time. For PPMPQS such a prediction formula is not known and the determination of an optimal choice of the parameters for a given number would require many full runs with that given number, and the use of an inadmissible amount of CPUtime. In order yet to provide measurements that can help to determine a good choice of the parameters in PPMPQS, we have factored many numbers in the 66  88 decimal digits range, where each number was run once with a specific choice of the parameters. In addition, an experimental prediction formula is given that has a restricted scope in the sense that it only applies to numbers of a given size, for a fixed choice of the parameters of PPMPQS. So such a formula may be useful if one wishes to factor many different large numbers of about the same size with PPMPQS."
Improved techniques for lower bounds for odd perfect numbers by
R. P Brent(
Book
)
6 editions published in 1989 in English and held by 9 WorldCat member libraries worldwide
Abstract: "If N is an odd perfect number, and q[superscript k] [symbol] N, q prime, k even, then it is almost immediate that N> q[superscript 2k]. We prove here that, subject to certain conditions verifiable in polynomial time, in fact N> q[superscript 5k/2]. Using this and related results, we are able to extend the computations in an earlier paper to show that N> 10[superscript 300]."
6 editions published in 1989 in English and held by 9 WorldCat member libraries worldwide
Abstract: "If N is an odd perfect number, and q[superscript k] [symbol] N, q prime, k even, then it is almost immediate that N> q[superscript 2k]. We prove here that, subject to certain conditions verifiable in polynomial time, in fact N> q[superscript 5k/2]. Using this and related results, we are able to extend the computations in an earlier paper to show that N> 10[superscript 300]."
Checking the Goldbach conjecture on a vector computer by
A Granville(
Book
)
4 editions published in 1988 in English and Undetermined and held by 8 WorldCat member libraries worldwide
"The Goldbach conjecture says that every even number can be expressed as the sum of two primes and it is known to be true up to 10⁸ (except for 2, if 1 is not considered a prime). This paper describes the results of a numerical verification of the Goldbach conjecture on a Cyber 205 vector computer up to the bound 2*10¹⁰. Some statistics and supporting results based on the Prime ktuplets conjecture of Hardy and Littlewood are presented."
4 editions published in 1988 in English and Undetermined and held by 8 WorldCat member libraries worldwide
"The Goldbach conjecture says that every even number can be expressed as the sum of two primes and it is known to be true up to 10⁸ (except for 2, if 1 is not considered a prime). This paper describes the results of a numerical verification of the Goldbach conjecture on a Cyber 205 vector computer up to the bound 2*10¹⁰. Some statistics and supporting results based on the Prime ktuplets conjecture of Hardy and Littlewood are presented."
Factoring with the quadratic sieve on large vector computers by
H. J. J. te Riele(
Book
)
4 editions published in 1988 in English and Undetermined and held by 7 WorldCat member libraries worldwide
Abstract: "The results are presented of experiments with the multiple polynomial version of the quadratic sieve factorization method on a Cyber 205 and on a NEC SX2 vector computer. Various numbers in the 5092 decimal digits range have been factorized, as a contribution to (i) the Cunningham project, (ii) Brent's Table of factors of Mersenne numbers, and (iii) a proof by Brent and G. Cohen of the nonexistence of odd perfect numbers below 10²⁰⁰. The factorized 92 decimal digits number is a record for general purpose factorization methods."
4 editions published in 1988 in English and Undetermined and held by 7 WorldCat member libraries worldwide
Abstract: "The results are presented of experiments with the multiple polynomial version of the quadratic sieve factorization method on a Cyber 205 and on a NEC SX2 vector computer. Various numbers in the 5092 decimal digits range have been factorized, as a contribution to (i) the Cunningham project, (ii) Brent's Table of factors of Mersenne numbers, and (iii) a proof by Brent and G. Cohen of the nonexistence of odd perfect numbers below 10²⁰⁰. The factorized 92 decimal digits number is a record for general purpose factorization methods."
Optimization of the MPQSfactoring algorithm on the Cyber 205 by
Walter M Lioen(
Book
)
2 editions published in 1988 in English and held by 7 WorldCat member libraries worldwide
2 editions published in 1988 in English and held by 7 WorldCat member libraries worldwide
On the zeros of the Riemann zeta function in the critical strip ; IV by
J. van de Lune(
Book
)
2 editions published in 1985 in English and held by 7 WorldCat member libraries worldwide
2 editions published in 1985 in English and held by 7 WorldCat member libraries worldwide
Optimization of the real level 2 BLAS on the Cyber 205 by
Walter M Lioen(
Book
)
4 editions published in 1987 in English and held by 7 WorldCat member libraries worldwide
4 editions published in 1987 in English and held by 7 WorldCat member libraries worldwide
Factorizations of Cunningham numbers with bases 13 to 99 : millennium edition by
R. P Brent(
Book
)
2 editions published in 2001 in English and held by 7 WorldCat member libraries worldwide
2 editions published in 2001 in English and held by 7 WorldCat member libraries worldwide
Computations concerning the conjecture of Mertens by
H. J. J. te Riele(
Book
)
4 editions published in 1978 in English and held by 7 WorldCat member libraries worldwide
4 editions published in 1978 in English and held by 7 WorldCat member libraries worldwide
On the sign of the difference pi(x)li(x) by
H. J. J. te Riele(
Book
)
2 editions published in 1986 in English and held by 7 WorldCat member libraries worldwide
2 editions published in 1986 in English and held by 7 WorldCat member libraries worldwide
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Related Identities
 Dekker, T. J. Other Editor
 Vorst, H. A. van der 1944 Other Editor
 Wiegerinck, Jan Editor
 Ran, A. C. M. (André C. M.) 1956 Editor
 Centrum voor Wiskunde en Informatica (Amsterdam, Netherlands)
 Lune, J. van de Author
 Brent, R. P. (Richard P.) Author
 Stichting Mathematisch centrum, Amsterdam
 Centrum voor Wiskunde en Informatica (Amsterdam, Netherlands) Afdeling Numerieke Wiskunde
 Lioen, W. M. Author
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Associated Subjects
Algebraic number theoryData processing Algebras, LinearData processing Algorithms Aliquot sequences Computer algorithms Computer programming Continued fractionsData processing CYBER 205 (Computer) Diophantine analysis Diophantine approximation Divisor theory Electronic digital computers Estimation theory Factorization (Mathematics) Factorization (Mathematics)Data processing Functions, Zeta Goldbach conjecture Integral equations Mathematical optimization Mathematics Möbius function Numbers, Prime Numbers, PrimeData processing Number theory Number theoryData processing Perfect numbers Search theory Semiconductors Sieves (Mathematics) Supercomputers Supercomputing Vector analysis Vector processing (Computer science)
Alternative Names
Herman te Riele Dutch mathematician
Herman te Riele niederländischer Mathematiker
Hermanus Johannes Joseph te Riele
Riele, H. J. J. te.
Riele, Herman te.
Riele, Herman te 1947
Riele, Hermanus Johannes Joseph te
Te Riele, H. J. J.
Te Riele, H. J. J. 1947
Te Riele, Herman
Te Riele Herman 1947....
Te Riele, Hermanus Johannes Joseph
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