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Enumerating Solutions to P(a) + Q(b) = R(c) + S(d)
, 1999
"... Let p; q; r; s be polynomials with integer coecients. This paper presents a fast method, using very little temporary storage, to nd all small integers (a; b; c; d) satisfying p(a)+q(b) = r(c)+s(d). Numerical results include all small solutions to a ; all small solutions to a ; ..."
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Let p; q; r; s be polynomials with integer coecients. This paper presents a fast method, using very little temporary storage, to nd all small integers (a; b; c; d) satisfying p(a)+q(b) = r(c)+s(d). Numerical results include all small solutions to a ; all small solutions to a ; and the smallest positive integer that can be written in 5 ways as a sum of two coprime cubes.
PLURALISM IN MATHEMATICS
, 2004
"... We defend pluralism in mathematics, and in particular Errett Bishop’s constructive approach to mathematics, on pragmatic grounds, avoiding the philosophical issues which have dissuaded many mathematicians from taking it seriously. We also explain the computational value of interval arithmetic. ..."
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We defend pluralism in mathematics, and in particular Errett Bishop’s constructive approach to mathematics, on pragmatic grounds, avoiding the philosophical issues which have dissuaded many mathematicians from taking it seriously. We also explain the computational value of interval arithmetic.
Can a Computer Proof be Elegant?
 SIGACT News 60(1), 111–114. JOACHIM VON ZUR GATHEN
, 2001
"... Introduction In computer science, proofs about computer algorithms are par for the course. Proofs by computer algorithms, on the other hand, are not so readily accepted. We present one viewpoint on computer assisted proofs, or what we call proofs via the computational method [13]. While one might e ..."
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Introduction In computer science, proofs about computer algorithms are par for the course. Proofs by computer algorithms, on the other hand, are not so readily accepted. We present one viewpoint on computer assisted proofs, or what we call proofs via the computational method [13]. While one might expect pure mathematicians to be a bit leery of the computational method, one would hope that computer scientists would be more receptive. After all, in seems contradictory for computer scientists not to believe in algorithms. In the author's experience, this has not been the case. The comment one often hears is \Can't you nd a better proof"? One of the arguments given against the computational method is that a computer assisted proof cannot possibly be elegant. We question this argument here. It is important that we dene precisely what we mean by the computational method. In the computational method we seek to
A Manifesto for the Computational Method
 In Fun with Algorithms: Proceedings of the International Conference
, 2000
"... We promote the much maligned computational method. The computational method is a paradigm for proving mathematical results where the burden of doing the \grunt work" is given to our able research assistant, the computer. We assert that proofs using the computational method, also known as computer ai ..."
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We promote the much maligned computational method. The computational method is a paradigm for proving mathematical results where the burden of doing the \grunt work" is given to our able research assistant, the computer. We assert that proofs using the computational method, also known as computer aided proofs, are here to stay. In fact, the use of a computer can make the analysis of complicated algorithms fun. We illustrate the usefulness of the method by analyzing a randomized algorithm for multiprocessor scheduling with rejection. More specically, we present a randomized algorithm which is 1.44127competitive. The best previously known result is a 1.5competitive algorithm. Key words: Computer assisted proofs, Online algorithms, Scheduling 1 What is the Computational Method? Throughout the ages, mathematicians have employed research assistants who perform the more menial tasks which might otherwise distract them from the main task at handproving results and thereby having fun. ...
Making the Analysis of Complicated Algorithms Fun: A Manifesto for the Computational Method
 In fun with Algorithms: Proceedings of the International Conference
, 1998
"... We promote the much maligned computational method. The computational method is a paradigm for proving mathematical results where the burden of doing the "grunt work" is given to our able research assistant, the computer. We assert that proofs using the computational method, also known as computer ai ..."
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We promote the much maligned computational method. The computational method is a paradigm for proving mathematical results where the burden of doing the "grunt work" is given to our able research assistant, the computer. We assert that proofs using the computational method, also known as computer aided proofs, are here to stay. In fact, the use of a computer can make the analysis of complicated algorithms fun. We illustrate the usefulness of the method by analyzing a randomized algorithm for multiprocessor scheduling with rejection. Keywords: Computer assisted proofs, Online algorithms, Scheduling. 1 What is the Computational Method? Throughout the ages, mathematicians have employed research assistants who perform the more menial tasks which might otherwise distract them from the main task at handproving results and thereby having fun. For instance, it is well known that Gauss employed a calculating prodigy who helped him in compiling trigonometric tables. Hardy had Ramanujanor...
Some Remarks and Problems in Number Theory Related to the Work of Euler
"... Motto. One is mathematics and Euler is its prophet. This phrase was coined half as a joke at a mathematical party in Budapest about 50 years ago by Tibor Gallai. In these remarks we mention some of the things the prophet Euler has handed down to us and sometimes give some later developments. Many of ..."
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Motto. One is mathematics and Euler is its prophet. This phrase was coined half as a joke at a mathematical party in Budapest about 50 years ago by Tibor Gallai. In these remarks we mention some of the things the prophet Euler has handed down to us and sometimes give some later developments. Many of the recollections and conjectures in these remarks are those of the first author, and first person references are used to keep the exposition informal. In 1737 Euler proved that the number of primes was infinite by showing that the sum of their reciprocals diverges, i.e., E i = oo. vp p He did this by using the (invalid) identity (1) 00 n=I n D t p Though invalidEuler rarely worried about convergenceit can be fixed by looking at n v as s 1. For this, see Ayoub [1], who said elsewhere [2] that Euler "laid the foundations of analytic number theory." Denote by 7T (X) the number of primes p < x. It is curious that Euler after having proved (1) never asked himself: how does 17(x) behave for large x? For (1) immediately implies that for infinitely many x, 1r(x)> x". In fact, for infinitely many x,1r(x)> x/(log x) ' +`. It seems to me that with a little experimentation Euler could have discovered the prime number theorem p lim 9(x) = 1. X oo X/log X After all, he did discover the quadratic reciprocity theorem by observation, and that seems to be at least as hard to see. But as we will see again later, such questions did not seem to occur to Euler. The prime number theorem was first conjectured shortly before Euler 's death by Legendre in 1780 in the form 1r (x) = x log Xc' with c 1.08. In 1792 Gauss, who was only 15 at the time, even noticed that 292 MATHEMATICS MAGAZINEdY _ r
SYMMETRIC HOMOGENEOUS DIOPHANTINE EQUATIONS OF ODD
, 809
"... Abstract. We find a parametric solution of an arbitrary symmetric homogeneous diophantine equation of 5th degree in 6 variables using two primitive solutions. We then generalize this approach to symmetric forms of any odd degree by proving the following results. (1) Every symmetric form of odd degre ..."
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Abstract. We find a parametric solution of an arbitrary symmetric homogeneous diophantine equation of 5th degree in 6 variables using two primitive solutions. We then generalize this approach to symmetric forms of any odd degree by proving the following results. (1) Every symmetric form of odd degree n ≥ 5 in 6 · 2 n−5 variables has a rational parametric solution depending on 2n − 8 parameters. (2) Let F(x1,..., xN) be a symmetric form of odd degree n ≥ 5 in N = 6 ·2 n−4 variables, and let q be any rational number. Then the equation F(xi) = q has a rational parametric solution depending on 2n − 6 parameters. The latter result can be viewed as a solution of a problem of Waring type for this class of forms. 1.