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Some remarks on computing the square parts of integers

Susan Landau
1988 Information and Computation  
We survey other recent work on computing the square part of an integer and give upper and lower bounds on the complexity of solving the problem.  ...  Let n be a positive integer, and suppose n = n pp' is its prime factorization. Let 0(n) = n pp-', so that n/0(n) is the largest squarefree factor of n.  ...  We begin our study of computing square parts of integers by surveying some related work. Suppose n is a positive integer with prime factorization n p:.  ... 
doi:10.1016/0890-5401(88)90028-4 fatcat:ucbxvq57pvb6ndywh3unaz366m

Farmer Ted Goes Natural [article]

Greg Martin
1999 arXiv   pre-print
It is shown that almost-squares can be equivalently described in a surprisingly elegant way, and that computing whether a number is an almost-square and computing the least almost-square not exceeding  ...  A traditional "Farmer Ted" calculus problem is to minimize the perimeter of a rectangular chicken coop given the area N, so that as little as possible will be spent on the fencing.  ...  The author would like to thank Andrew Granville and the anonymous referees for their valuable comments which improved the presentation of this paper.  ... 
arXiv:math/9807108v2 fatcat:3ekukl4wazh3xj5bvmprpofmeu

Integration of rational functions: Rational computation of the logarithmic part

D Lazard, R Rioboo
1990 Journal of symbolic computation  
A new formula is given for the logarithmic part of the integral of a rational function, one that strongly improves previous algorithms and does not need any computation in an algebraic extension of the  ...  field of constants, nor any factorisation since only polynomial arithmetic and GCD computations are used.  ...  But the computation of (2) needs several gcd calculations in algebraic extensions, whereas in fact Rb does not depend on the value of b but only on the minimal polynomial of b.  ... 
doi:10.1016/s0747-7171(08)80026-0 fatcat:bzivprvwrbfmph7mj3oqhmxkwi

New recurrences for Euler''s partition function

Mircea MERCA
2017 Turkish Journal of Mathematics  
As a corollary of these results, we obtain an efficient method to compute the parity of Euler's partition function p(n) that requires only the parity of p(k) with k n/4.  ...  In this paper, the author invokes some consequences of the bisectional pentagonal number theorem to derive two linear recurrence relations for Euler's partition function p(n).  ...  Acknowledgement The author likes to thank the referees for their helpful comments. Special thanks go to Dr. Oana Merca for the careful reading of the manuscript and helpful remarks.  ... 
doi:10.3906/mat-1604-124 fatcat:kgtdr53bd5gtlkliqhxmdr6h4a

Problems on combinatorial properties of primes [article]

Zhi-Wei Sun
2016 arXiv   pre-print
For example, we conjecture that for any integer n>1 one of the n numbers π(n),π(2n),...  ...  One of our conjectures involving the partition function p(n) states that for any prime p there is a primitive root g<p modulo p with g∈{p(n): n=1,2,3,...}.  ...  Then, π((k + 1)n) − π(kn) (the number of primes in the interval (kn, (k + 1)n]) is a square for some k = 0, . . . , n − 1. Remark 2.7.  ... 
arXiv:1402.6641v12 fatcat:m5o5wxr23jg5vhbwmtgt5q55vm

Farmer Ted Goes Natural

Greg Martin
1999 Mathematics Magazine  
The author would like to thank Andrew Granville and the anonymous referees for their valuable comments which improved the presentation of this paper.  ...  The author would also like to acknowledge the support of National Science Foundation grant number DMS 9304580 . . . the NSF may or may not wish to acknowledge this paper.  ...  This shows that A(N ) can be computed in polynomial time, which establishes part (c) of the corollary. Suppose now that we want to compute the N th almost-square.  ... 
doi:10.1080/0025570x.1999.11996746 fatcat:mgj2cfdavzcdjmzrnvfajs4uzq

Farmer Ted Goes Natural

Greg Martin
1999 Mathematics Magazine  
The author would like to thank Andrew Granville and the anonymous referees for their valuable comments which improved the presentation of this paper.  ...  The author would also like to acknowledge the support of National Science Foundation grant number DMS 9304580 . . . the NSF may or may not wish to acknowledge this paper.  ...  This shows that A(N ) can be computed in polynomial time, which establishes part (c) of the corollary. Suppose now that we want to compute the N th almost-square.  ... 
doi:10.2307/2691219 fatcat:zvacvlawjnabva2up42w4napki

PROBLEMS ON COMBINATORIAL PROPERTIES OF PRIMES

ZHI-WEI SUN
2015 Number Theory: Plowing and Starring Through High Wave Forms  
To convince oneself, one has only to glance at the tables of primes which some people took the trouble to computer beyond a hundred thousand, and one perceives that there is no order and no rule.  ...  We will also mention our computational evidence to support the related conjectures. / 64 Part II. Combinatorial properties of π(x) 9 / 64  ...  We defineq(n) = p(n) − q(n), which is the number of partitions of n with some part repeated (or even).  ... 
doi:10.1142/9789814644938_0006 fatcat:lninta44rremhfocxhwg4cbibi

Fast Computation Algorithm for Discrete Resonances among Gravity Waves

Elena Kartashova
2006 Journal of Low Temperature Physics  
Numerical simulations of the last few years showed unambiguously the existence of some discrete effects in the short-waves part of the wave spectrum.  ...  Example of such an algorithm for 4-waves interactions of gravity waves is given. Its generalization on the different types of waves is briefly discussed.  ...  2 , Lagrange and Euler theorems should be replaced by some known numbertheoretical results on the decomposition of an integer into the sum of three squares.  ... 
doi:10.1007/s10909-006-9237-1 fatcat:lkgb2lynhbd37nqrmifqnxar7u

Prime factorization using square root approximation

Joseph Zalaket, Joseph Hajj-Boutros
2011 Computers and Mathematics with Applications  
The security of RSA relies on the difficulty of factoring large integers.  ...  Many cryptosystems are based on the factorization of large integers.  ...  We propose a new heuristic method based on the square root approximation that allows factoring large integers.  ... 
doi:10.1016/j.camwa.2011.02.027 fatcat:hdjphnv6hncl7k5niuwwuyroae

Implementation of a new primality test

H. Cohen, A. K. Lenstra
1987 Mathematics of Computation  
An implementation of the Cohen-Lenstra version of the Adleman-Pomerance-Rumely primality test is presented.  ...  Primality of prime numbers of up to 213 decimal digits can now routinely be proved within approximately ten minutes.  ...  Winter for writing the multiprecision routines. Université de Bordeaux I Talence, France  ... 
doi:10.1090/s0025-5718-1987-0866102-2 fatcat:5xcu4xzikbal3d4allxc5qtf7a

Implementation of a New Primality Test

H. Cohen, A. K. Lenstra
1987 Mathematics of Computation  
An implementation of the Cohen-Lenstra version of the Adleman-Pomerance-Rumely primality test is presented.  ...  Primality of prime numbers of up to 213 decimal digits can now routinely be proved within approximately ten minutes.  ...  Winter for writing the multiprecision routines. Université de Bordeaux I Talence, France  ... 
doi:10.2307/2007877 fatcat:wg2fbdrmcvac7pnpu23ej6k3zi

ON A DIOPHANTINE EQUATION OF CASSELS

F. LUCA, P. G. WALSH
2005 Glasgow Mathematical Journal  
Cassels gave a solution to the problem of determining all instances of the sum of three consecutive cubes being a square.  ...  This amounts to finding all integer solutions to the Diophantine equation y 2 = 3x(x 2 + 2).  ...  The first author was supported in part by the grant SEP-CONACyT 37259E. The second author gratefully acknowledges support from the Natural Sciences and Engineering Research Council of Canada.  ... 
doi:10.1017/s001708950500251x fatcat:pyv7jpdgnjegtatf7faryuzxvq

Primes in the denominators of Igusa Class Polynomials [article]

Kristin Lauter
2003 arXiv   pre-print
are bounded by d, the absolute value of the discriminant of K, and 2) that they divide d-x^2, for some integer x whose square is less than d.  ...  The purpose of this note is to suggest an analogue for genus 2 curves of part of Gross and Zagier's work on elliptic curves.  ...  are bounded by d, the absolute value of the discriminant of K, and 2) that they divide d − x 2 , for some integer x whose square is less than d.  ... 
arXiv:math/0301240v1 fatcat:ug4ityysebavlgpuf7hxcsmo3u

Page 349 of American Mathematical Society. Bulletin of the American Mathematical Society Vol. 29, Issue [page]

1923 American Mathematical Society. Bulletin of the American Mathematical Society  
In seeking the relations thus suggested, we find at the outset some remarkable types of congruences which deserve independent notice on account of their generality.  ...  r is prime to p, and one of the following: the familiar denumerants of the classical theory of partitions; two new functions depending upon those partitions of an integer in which no part appears more  ... 
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