Prof. Gary Walsh - Research Articles

Prof. Gary Walsh

Research

via AMS:

mathscinet-ams-org.proxy.bib.uottawa.ca/mathscinet/author?authorId=230332

Publication List (in html format)

P.G. Walsh, An observation concerning the rank of $y^2=x^3+2nx$. to appear in

<a href="https://mathsociety.ph/matimyas/">Matimyas Mat.<a> 2025.

P.G. Walsh, A question of Erdos on 3-powerful numbers and an elliptic curve analogue of the

Ankeny-Artin-Chowla conjecture. to appear in

<a href="https://web.math.pmf.unizg.hr/~duje/radhazumz/prihvaceni.html">Rad Hazu.<a> 2025.

P.G. Walsh, Explicit estimates on a theorem of Shioda concerning the ranks of curves given by

$y^2=x^3-a^2x+m^2. to appear in

<a href="http://www.math.klte.hu/publi">Publicationes Mathematicae Debrecen<a> 2025.

P.G. Walsh, Integral Points on Elliptic Curves: An Exploration into Speculative Number Theory.

to appear in <a href="https://dl.acm.org/toc/sigact/2023/">SIGACT News.</a>, 20 pages, 2024.

M.A. Bennett and P.G. Walsh, Computations and solutions to a problem of Erdos concerning four coprime powerful numbers in arithmetic progression. 2023 INTEGERS conference,

<a href="https://math.colgate.edu/~integers/a3Proc23/a3Proc23.pdf2024">INTEGERS</a>.

P.G. Walsh, Specializations of a generic rank 2 curve of Shioda. to appear in <a href="https://www.sciencedirect.com/journal/journal-of-number-theory">J. Number Theory.</a>, 2023.

P.G. Walsh, Squares in recurrences using elliptic curves.

to appear in <a href="https://www.worldscientific.com/worldscinet/ijnt">International J. Number Theory.</a> (2023)

P.G. Walsh, A note on lower bounds for ranks using Pell equations. <a href="https://web.math.pmf.unizg.hr/~duje/radhazumz/">Rad Hrvat. Akad. Znan. Umjet. Mat. Znan.</a> vol. 27, 2023.

P.G. Walsh

Observations concerning the representation of positive integers as a sum of three cubes, (2023)

to appear in <a href="https://rmmc.asu.edu/rmj/rmj.html">Rocky Mountain Journal of Mathematics</a>

J. Grantham and P.G. Walsh, Representing integers as a sum of three cubes.

see NT arxiv Nov. 2022. pari code <a href="https://mysite.science.uottawa.ca/gwalsh/sumcubes10.txt">here.</a>

P.G. Walsh,

The trace of Frobenius for curves of the form $y^2=x^3+dx$.

2022, to appear in <a href="https://ami.uni-eszterhazy.hu/">Annales Mathematicae et Informaticae</a>

P.G. Walsh,

An effective version of a theorem of Shioda on the rank of an elliptic curve given by $y^2=f(x)+m^2$,

<a href="http://math.colgate.edu/~integers/">Integers Journal.</a> 22 , (2022).

A. Togbe and P.G. Walsh,

A classical approach to a parametric family of simultaneous Pell equations with applications to a family of Thue equations.

<a href="https://www.springer.com/journal/40590">Boletin de la Sociedad Matemática Mexicana</a>. 28 , (2022).

M.A. Bennett and P.G. Walsh,

A note on the Computation of Integral Bases in Pure Quartic Number Fields.

<a href="http://www.math.klte.hu/publi">Publicationes Mathematicae Debrecen</a> 100 , (2022).

P.G. Walsh,

Corrigendum on "On two classes of simultaneous Pell equations with no solutions".

<a href="http://www.ams.org/mcom">Math. Comp.</a> 90 (2021), 2503-2505.

N. Lin, P.G. Walsh and Ping-Zhi Yuan,

Sharp bounds for the number of integral points on

$y^2 = x^3 \pm tx^2 + tpx$.

Publ. Math. Debrecen 98 (3/4), (2021).

<a href="http://publi.math.unideb.hu/load_jpg.php?p=2444">Publicationes Mathematicae Debrecen</a>

<!--- P.G. Walsh, Pingzhi Yuan, and Z. Zhang,

The diophantine equation $x^2-(4p^{2m}+1)y^4=-4p^{2m}$ II, submitted. --->

B. He, A. Togbe and P.G. Walsh,

On the intersection of Lucas sequences of distinct type, <a href="http://www.labmath.uqam.ca/~annales/">Ann. Sci. Math. Quebec</a> 35 (2011), 31-61.

P.G. Walsh,

Maximal ranks and integer points on a family of elliptic curves II,

<a href="http://rmmc.eas.asu.edu/abstracts/rmj/rmj.htm">Rocky Mountain Journal of Mathematics</a>

41 (2011), 311-317.

Yang Hai and P.G. Walsh

On a Diophantine problem of Bennett,

<a href="http://journals.impan.gov.pl/aa/">Acta Arithmetica</a> 145 (2010), 129-136.

Michael Stoll, P.G. Walsh and Pingzhi Yuan,

The diophantine equation $x^2-(2^{2m}+1)y^4=-2^{2m}$ II,

<a href="http://journals.impan.gov.pl/aa/">Acta Arithmetica</a> 139 (2009), 57-63.

P.G. Walsh,

On the number of large integer points on elliptic curves.

<a href="http://journals.impan.gov.pl/aa/">Acta Arithmetica</a> 138 (2009), 317-327.

P.G. Walsh,

Maximal ranks and integer points on a family of elliptic curves.

<a href="http://www.math.hr/glasnik/index.html">Glasnik Matematicki</a> 44 (2009), 83-87.

F. Luca and P.G. Walsh,

On a sequence of integers arising from a system of Pell equations,

<a href="http://www.staff.amu.edu.pl/~fa/">Functiones et Approximatio Commentarii Mathematici</a>

38 (2008), 111-116.

B. He, A. Togbe and P.G. Walsh,

The diophantine equation $x^2-(2^{2m}+1)y^4=-2^{2m}$,

<a href="http://www.math.klte.hu/publi">Publicationes Mathematicae Debrecen</a>

73 (2008).

P.G. Walsh,

The integer solutions to $y^2=x^3 \pm p^k x$,

<a href="http://rmmc.eas.asu.edu/abstracts/rmj/rmj.htm">Rocky Mountain Journal of Mathematics</a>

38 (2008), 1285-1301.

S. Akhtari, A. Togbe and P.G. Walsh

The Diophantine equation aX^4-bY^2=2,

<a href="http://journals.impan.gov.pl/aa/">Acta Arithmetica</a>

131 (2008), 145-169.

Addendum: <a href="http://journals.impan.gov.pl/aa/">Acta Arithmetica</a>

137 (2009), 199-202.

P.G. Walsh,

On a question of Kaplansky. II.,

<a href="http://www.albmath.org/">Albanian J. Math.</a>

2 no. 1, March 2008, 3-5.

F. Luca and P.G. Walsh,

On a diophantine equation related to a conjecture of Erdos and Graham,

<a href="http://www.math.hr/glasnik/index.html">Glasnik Matematicki</a>

42 (2007), 281-289.

P.G. Walsh,

Sharp bounds for the number of solutions to simultaneous Pell equations,

<a href="http://journals.impan.gov.pl/aa/">Acta Arithmetica</a>

126 (2007), 125-137.

P.G. Walsh,

On a very special class of Ramanujan-Nagell type Diophantine equations,

to appear in <a href="http://www.pphmj.com/abstract/2132.htm">Far East J. Math.</a> 24 no.1 (2007), 55-58.

M.A. Bennett, A. Togbe, and P.G. Walsh,

A generalization of a theorem of Bumby,

<a href="http://www.worldscinet.com/ijnt/ijnt.shtml">International Journal of Number Theory</a> 2 no. 2 (2006), 195-206.

P.G. Walsh,

On a family of quartic equations and an elementary solution to a Diophantine problem of Martin Gardner,

<a href="http://www.math.hr/glasnik/index.html">Glasnik Matematicki</a> 41 (2006), 217-221.

A. Dujella, C. Fuchs and P.G. Walsh, Diophantine m-tuples for linear polynomials II,

<a href="http://www.sciencedirect.com/science?_ob=JournalURL&_cdi=6904&_auth=y&_acct=C000028338&_version=1&_urlVersion=0&_userid=554534&md5=1b389dff774d57adfc63745c2001e144">Journal of Number Theory</a> 120 (2006), 213-228.

D. Poulakis and P.G. Walsh, A note on the Diophantine equation $x^2-dy^4=1$ with prime discrminant. II,

<a href="http://journals.impan.gov.pl/cm/">Colloquium Mathematicum.</a> 105 (2006), 51-55.

F. Luca, C.F. Osgood and P.G. Walsh, Diophantine approximations and a problem from the

1988 IMO,<a href="http://rmmc.eas.asu.edu/abstracts/rmj/rmj.htm">Rocky Mountain Journal of Mathematics</a>

36 (2006), 637-648.

A. Togbe, P.M. Voutier, and P.G. Walsh,

Solving a family of Thue equations with an application to the equation $x^2-dy^4=1$,

<a href="http://journals.impan.gov.pl/aa/">Acta Arithmetica</a>

120 (2005), 39-58.

F. Luca and P.G. Walsh, On a Diophantine equation of Cassels,

<a href="http://www.cambridge.org/uk/journals/journal_catalogue.asp?historylinks=ALPHA&mnemonic=GMJ">Glasgow

Mathematical Journal</a> 47 (2005), 303-307.

P.G. Walsh, Squares in Lucas sequences with rational roots,

<a href="http://www.integers-ejcnt.org">INTEGERS: The Electronic Journal of Combinatorial Number Theory</a>

5 no.3 (2005) A15.

D. Poulakis and P.G. Walsh, A note on the Diophantine equation $x^2-dy^4=1$ with prime discrminant,

<a href="http://comptes.math.carleton.ca/">Comptes Rendues Math. Sci. Canada</a> 27 no. 2 (2005), 54-57.

P.G. Walsh, A note on class number one criteria of Sirola for real quadratic fields,

<a href="http://www.math.hr/glasnik/index.html">Glasnik Matematicki</a> 40 (2005), 21-27.

F. Luca and P.G. Walsh, On the number of nonquadratic residues which are not primitive

roots, <a href="http://journals.impan.gov.pl/cm/">Colloquium Mathematicum.</a> 100 (2004), 91-93.

P.G. Walsh, Diophantine applications of Bennett's abc theorem,

<a href="http://www.math.klte.hu/publi">Publicationes Mathematicae Debrecen.</a> 65 (2004), 497-512.

E. Herrmann and P.G. Walsh, On the determination of values in certain ternary recurrence sequences and

the rational torsion on a family of elliptic curves arising from the work of H.C. Williams,

In <a href="http://www.fields.utoronto.ca/programs/scientific/02-03/numtheory/">High Primes and Misdemeanours,</a>

Lectures in Honour of the 60th Birthday of Hugh Cowie Williams, Fields Institute Communications

41 (2004), 227-235. A.J. van der Poorten and A. Stein Ed's.

E. Herrmann, F. Luca, P.G. Walsh, A note on the Ramanujan-Nagell equation,

<a href="http://www.math.klte.hu/publi">Publicationes Mathematicae Debrecen.</a> 64 (2004), 21-30.

M.J. Jacobson, Jr., A. Pinter, P.G. Walsh, A computational

method for solving the Diophantine equation $y^2=1^k+2^k+...+x^k$,

<a href="http://www.ams.org/mcom">Mathematics of Computation</a> 72 (2003), 2099-2110.

P.G. Walsh, Near Squares in linear recurrence sequences,

<a href="http://www.math.hr/glasnik/index.html">Glasnik Matematicki</a> 38 no.1 (2003), 11-18.

P.G. Walsh, On subsums of units in pure cubic number fields,

<a href="http://comptes.math.carleton.ca/">Comptes Rendues Math. Sci. Canada</a> 25 no. 1 (2003).

F.Luca and P.G. Walsh, The product of like-indexed terms

in binary recurrences,

96 (2002), 152-173.

P.G. Walsh, On a question of Kaplansky,

<a href="http://www.maa.org/pubs/monthly.html">The American Mathematical Monthly</a>

109 August-September 2002, 660-661.

P.G. Walsh, An improved method for solving the Thue equations

$X^4-2rX^2Y^2-sY^4=1$, in

<a href="http://www.akpeters.com/book.asp?bID=161">Number Theory for the Millennium I</a>

(Proceedings of the Millennial Number Theory Conference, Champaign-Urbana,

Illinois, May 2000), Bruce Berndt, et.al., editors, A.K. Peters Ltd, 2002.

F. Luca and P.G. Walsh, Squares in Lehmer sequences with

Diophantine applications,

<a href="http://journals.impan.gov.pl/aa/">Acta Arithmetica</a>

100 (2001), 47-62.

F. Luca and P.G. Walsh, A generalization of a theorem of Cohn on

the equations $x^3-Ny^2= \pm 1$,

<a href="http://rmmc.eas.asu.edu/abstracts/rmj/rmj.htm">Rocky Mountain Journal of Mathematics</a>

31 no. 2 (2001), 503-509.

P.G. Walsh, A Polynomial-time complexity bound for the computation

of the singular part of a Puiseux expansion of an algebraic function,

<a href="http://www.ams.org/mcom">Mathematics of Computation</a>

69 (2000), 1167-1182.

P.G. Walsh, Irreducibility testing over local fields,

<a href="http://www.ams.org/mcom">Mathematics of Computation</a>

69 (2000), 1183-1193.

M.A. Bennett and P.G. Walsh, Simultaneous Pell equations with few

or no solutions,

Indagationes Mathematicae</a>

11 (2000), 1-12.

P.G. Walsh, Diophantine equations of the form

$aX^4-bY^2= \pm 1$, in

<a href="http://www.degruyter.de/catalog/1520.html">Algebraic Number Theory and

Diophantine Analysis.</a>

(Proceedings of an ICM satellite conference in

Graz, Austria, August 30 to September 5, 1998),

F. Halter-Koch and R. Tichy, editors, Walter deGruyter, 2000.

P.G. Walsh, On Diophantine equations of the form

$(x^m-1)(y^n-1)=z^2$, in

<a href="http://tatra.mat.savba.sk/">Tatra Mt. Math. Publ.</a>

20 (2000), 1-3.

P.G. Walsh, A note on a theorem of Ljunggren and the Diophantine

equations $x^2-kxy^2+y^4=1,4$,

Archive der Mathematik</a> 72 (1999), 1-7.

P.G. Walsh, The Diophantine equation

$X^2-db^2Y^4=1$,

Acta Arithmetica</a>

57 (1999), 179-188.

P. Ribenboim and P.G. Walsh, The ABC conjecture and the

Powerful part of terms in binary recurring sequences.

74 (1999), 134-147.

P.G. Walsh, Efficiency vs. Security in the implementation of

Public-Key Cryptography Proceedings of the 1999 Australasian Conference

on Theoretical Computer Science and Discrete Mathematics, 81-105.

M.A. Bennett and P.G. Walsh, The Diophantine equation

$b^2X^4-dY^2=1$.

<a href="http://www.ams.org/proc">Proceedings of the AMS</a>

127 (1999).

P.G. Walsh, On the complexity of rational Puiseux expansions.

<a href="http://nyjm.albany.edu:8000/PacJ/Papers.html">Pacific Journal

of Mathematics</a>

188 (1999), 369-387.

A.J. van der Poorten and P.G. Walsh, A note on Jacobi symbols

and continued fractions,

<a href="http://www.maa.org/pubs/monthly.html">American Mathematical

Monthly</a>

106 no. 1 (1999), 52-56.

P.G. Walsh, On a conjecture of Schinzel and Tijdeman.

in <a href="http://www.degruyter.de/catalog/1520.html">Number Theory in Progress</a>

p. 577-582. (proceedings of a conference in honour of the sixtieth

birthday of A. Schinzel, Zakopane, Poland, 1997) Walter de Gruyter, 1999.

P.G. Walsh, A polynomial complexity bound for computations on curves.

<a href="http://www.siam.org/journals/sicomp/sicomp.htm">SIAM Journal on Computing</a>

28 (1999), 704-708.

P.G. Walsh, Two classes of simultaneous Pell equations with no solutions.

<a href="http://www.ams.org/mcom">Mathematics of Computation</a>

68 (1999), 385-388.

P.G. Walsh, A note on Ljunggren's theorem about the Diophantine

equation $aX^2-bY^4=1$.

<a href="http://comptes.math.carleton.ca/">Comptes Rendues</a>

Mathematical Reports of the

Royal Society of Canada 20 (1998), 113-119.

P.G. Walsh, On integer solutions to $x^2-dy^2=1,z^2-2dy^2=1$.

<a href="http://journals.impan.gov.pl/aa/">Acta Arithmetica</a>

87 (1997), 69-76.

P.G.Walsh, A quantitative version of Runge's theorem on Diophantine

equations.

<a href="http://journals.impan.gov.pl/aa/">Acta Arithmetica</a>

62 (1992), 157-172.

<H3>Summer Undergraduate Research Papers</H3>

R.A.Mollin and P.G.Walsh, Proper differences of non-square powerful

numbers.

<a href="http://comptes.math.carleton.ca/">Comptes Rendues</a>

Mathematical Reports of the Royal Society of Canada 10 no. 2 (1988), 71-76.

R.A.Mollin and P.G.Walsh, On unit solutions of the equation

$xyz=x+y+z$ in the ring of integers of a quadratic field.

<a href="http://journals.impan.gov.pl/aa/">Acta Arithmetica</a>

48 (1987), 341-345.

R.A.Mollin and P.G.Walsh, On non-square powerful numbers.

<a href="http://www.sdstate.edu/~wcsc/http/fibcurrent.html">Fibonacci Quarterly</a>

25 (1987), 34-37.

R.A.Mollin and P.G.Walsh, On powerful numbers.

<a href="http://www.math.helsinki.fi/EMIS/journals/IJMMS/index.html">International J. Math. and Math. Sci.</a> {\bf 9} (1986), 801-806.

R.A.Mollin and P.G.Walsh, A note on quadratic fields,

powerful numbers, and the Pellian.

<a href="http://www.math.carleton.ca/comptes/comptes.html">Comptes Rendues</a>

Mathematical Reports of the Royal Society of Canada

8

no. 2 (1986), 109-114.

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