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Some remarks on computing the square parts of integers

1988
*
Information and Computation
*

We survey other recent work

doi:10.1016/0890-5401(88)90028-4
fatcat:ucbxvq57pvb6ndywh3unaz366m
*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:. ...##
###
Farmer Ted Goes Natural
[article]

1999
*
arXiv
*
pre-print

It is shown that almost-

arXiv:math/9807108v2
fatcat:3ekukl4wazh3xj5bvmprpofmeu
*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. ...##
###
Integration of rational functions: Rational computation of the logarithmic part

1990
*
Journal of symbolic computation
*

A new formula is given for

doi:10.1016/s0747-7171(08)80026-0
fatcat:bzivprvwrbfmph7mj3oqhmxkwi
*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. ...##
###
New recurrences for Euler''s partition function

2017
*
Turkish Journal of Mathematics
*

As a corollary

doi:10.3906/mat-1604-124
fatcat:kgtdr53bd5gtlkliqhxmdr6h4a
*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*. ...##
###
Problems on combinatorial properties of primes
[article]

2016
*
arXiv
*
pre-print

For example, we conjecture that for any

arXiv:1402.6641v12
fatcat:m5o5wxr23jg5vhbwmtgt5q55vm
*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. ...##
###
Farmer Ted Goes Natural

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*. ...

##
###
Farmer Ted Goes Natural

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*. ...

##
###
PROBLEMS ON COMBINATORIAL PROPERTIES OF PRIMES

2015
*
Number Theory: Plowing and Starring Through High Wave Forms
*

To convince oneself,

doi:10.1142/9789814644938_0006
fatcat:lninta44rremhfocxhwg4cbibi
*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). ...##
###
Fast Computation Algorithm for Discrete Resonances among Gravity Waves

2006
*
Journal of Low Temperature Physics
*

Numerical simulations

doi:10.1007/s10909-006-9237-1
fatcat:lkgb2lynhbd37nqrmifqnxar7u
*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*. ...##
###
Prime factorization using square root approximation

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*. ...

##
###
Implementation of a new primality test

1987
*
Mathematics of Computation
*

An implementation

doi:10.1090/s0025-5718-1987-0866102-2
fatcat:5xcu4xzikbal3d4allxc5qtf7a
*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 ...##
###
Implementation of a New Primality Test

1987
*
Mathematics of Computation
*

An implementation

doi:10.2307/2007877
fatcat:wg2fbdrmcvac7pnpu23ej6k3zi
*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 ...##
###
ON A DIOPHANTINE EQUATION OF CASSELS

2005
*
Glasgow Mathematical Journal
*

Cassels gave a solution to

doi:10.1017/s001708950500251x
fatcat:pyv7jpdgnjegtatf7faryuzxvq
*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. ...##
###
Primes in the denominators of Igusa Class Polynomials
[article]

2003
*
arXiv
*
pre-print

are bounded by d,

arXiv:math/0301240v1
fatcat:ug4ityysebavlgpuf7hxcsmo3u
*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. ...##
###
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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