Gary Walsh's Publication List

Research Articles

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 Rad Hazu. 2024.

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 Publicationes Mathematicae Debrecen 2024.

P.G. Walsh, Integral Points on Elliptic Curves: An Exploration into Speculative Number Theory. to appear in SIGACT News., 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, INTEGERS.

P.G. Walsh, Specializations of a generic rank 2 curve of Shioda. to appear in J. Number Theory., 2023.

P.G. Walsh, Squares in recurrences using elliptic curves. to appear in International J. Number Theory. (2023)

P.G. Walsh, A note on lower bounds for ranks using Pell equations. Rad Hrvat. Akad. Znan. Umjet. Mat. Znan. vol. 27, 2023.

P.G. Walsh Observations concerning the representation of positive integers as a sum of three cubes, (2023) to appear in Rocky Mountain Journal of Mathematics

J. Grantham and P.G. Walsh, Representing integers as a sum of three cubes. see NT arxiv Nov. 2022. pari code here.

P.G. Walsh, The trace of Frobenius for curves of the form $y^2=x^3+dx$. 2022, to appear in Annales Mathematicae et Informaticae

P.G. Walsh, An effective version of a theorem of Shioda on the rank of an elliptic curve given by $y^2=f(x)+m^2$, Integers Journal. 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. Boletin de la Sociedad Matemática Mexicana. 28 , (2022).

M.A. Bennett and P.G. Walsh, A note on the Computation of Integral Bases in Pure Quartic Number Fields. Publicationes Mathematicae Debrecen 100 , (2022).

P.G. Walsh, Corrigendum on "On two classes of simultaneous Pell equations with no solutions". Math. Comp. 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). Publicationes Mathematicae Debrecen

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B. He, A. Togbe and P.G. Walsh, On the intersection of Lucas sequences of distinct type, Ann. Sci. Math. Quebec 35 (2011), 31-61.

P.G. Walsh, Maximal ranks and integer points on a family of elliptic curves II, Rocky Mountain Journal of Mathematics 41 (2011), 311-317.

Yang Hai and P.G. Walsh On a Diophantine problem of Bennett, Acta Arithmetica 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, Acta Arithmetica 139 (2009), 57-63. (pdf)

P.G. Walsh, On the number of large integer points on elliptic curves. Acta Arithmetica 138 (2009), 317-327. (pdf)

P.G. Walsh, Maximal ranks and integer points on a family of elliptic curves. Glasnik Matematicki 44 (2009), 83-87. (pdf)

F. Luca and P.G. Walsh, On a sequence of integers arising from a system of Pell equations, Functiones et Approximatio Commentarii Mathematici 38 (2008), 111-116. (pdf)

B. He, A. Togbe and P.G. Walsh, The diophantine equation $x^2-(2^{2m}+1)y^4=-2^{2m}$, Publicationes Mathematicae Debrecen 73 (2008). (pdf)

P.G. Walsh, The integer solutions to $y^2=x^3 \pm p^k x$, Rocky Mountain Journal of Mathematics 38 (2008), 1285-1301. (pdf)

S. Akhtari, A. Togbe and P.G. Walsh The Diophantine equation aX^4-bY^2=2, Acta Arithmetica 131 (2008), 145-169. (pdf)
Addendum: Acta Arithmetica 137 (2009), 199-202. (pdf)

P.G. Walsh, On a question of Kaplansky. II., Albanian J. Math. 2 no. 1, March 2008, 3-5. (pdf)

F. Luca and P.G. Walsh, On a diophantine equation related to a conjecture of Erdos and Graham, Glasnik Matematicki 42 (2007), 281-289. (pdf)

P.G. Walsh, Sharp bounds for the number of solutions to simultaneous Pell equations, Acta Arithmetica 126 (2007), 125-137. (pdf)

P.G. Walsh, On a very special class of Ramanujan-Nagell type Diophantine equations, to appear in Far East J. Math. 24 no.1 (2007), 55-58. (pdf)

M.A. Bennett, A. Togbe, and P.G. Walsh, A generalization of a theorem of Bumby, International Journal of Number Theory 2 no. 2 (2006), 195-206. (pdf)

P.G. Walsh, On a family of quartic equations and an elementary solution to a Diophantine problem of Martin Gardner, Glasnik Matematicki 41 (2006), 217-221. (pdf)

A. Dujella, C. Fuchs and P.G. Walsh, Diophantine m-tuples for linear polynomials II, Journal of Number Theory 120 (2006), 213-228. (pdf)

D. Poulakis and P.G. Walsh, A note on the Diophantine equation $x^2-dy^4=1$ with prime discrminant. II, Colloquium Mathematicum. 105 (2006), 51-55. (pdf)

F. Luca, C.F. Osgood and P.G. Walsh, Diophantine approximations and a problem from the 1988 IMO,Rocky Mountain Journal of Mathematics 36 (2006), 637-648. (pdf)

A. Togbe, P.M. Voutier, and P.G. Walsh, Solving a family of Thue equations with an application to the equation $x^2-dy^4=1$, Acta Arithmetica 120 (2005), 39-58. (pdf)

F. Luca and P.G. Walsh, On a Diophantine equation of Cassels, Glasgow Mathematical Journal 47 (2005), 303-307. (pdf)

P.G. Walsh, Squares in Lucas sequences with rational roots, INTEGERS: The Electronic Journal of Combinatorial Number Theory 5 no.3 (2005) A15. (pdf)

D. Poulakis and P.G. Walsh, A note on the Diophantine equation $x^2-dy^4=1$ with prime discrminant, Comptes Rendues Math. Sci. Canada 27 no. 2 (2005), 54-57. (pdf)

P.G. Walsh, A note on class number one criteria of Sirola for real quadratic fields, Glasnik Matematicki 40 (2005), 21-27. (pdf)

F. Luca and P.G. Walsh, On the number of nonquadratic residues which are not primitive roots, Colloquium Mathematicum. 100 (2004), 91-93. (pdf)

P.G. Walsh, Diophantine applications of Bennett's abc theorem, Publicationes Mathematicae Debrecen. 65 (2004), 497-512. (pdf)

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 High Primes and Misdemeanours, 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. (pdf)

E. Herrmann, F. Luca, P.G. Walsh, A note on the Ramanujan-Nagell equation, Publicationes Mathematicae Debrecen. 64 (2004), 21-30. (pdf)

M.J. Jacobson, Jr., A. Pinter, P.G. Walsh, A computational method for solving the Diophantine equation $y^2=1^k+2^k+...+x^k$, Mathematics of Computation 72 (2003), 2099-2110. (pdf)

P.G. Walsh, Near Squares in linear recurrence sequences, Glasnik Matematicki 38 no.1 (2003), 11-18. (pdf)

P.G. Walsh, On subsums of units in pure cubic number fields, Comptes Rendues Math. Sci. Canada 25 no. 1 (2003).

F.Luca and P.G. Walsh, The product of like-indexed terms in binary recurrences, Journal of Number Theory 96 (2002), 152-173. (pdf)

P.G. Walsh, On a question of Kaplansky, The American Mathematical Monthly 109 August-September 2002, 660-661. (pdf)

P.G. Walsh, An improved method for solving the Thue equations $X^4-2rX^2Y^2-sY^4=1$, in Number Theory for the Millennium I (Proceedings of the Millennial Number Theory Conference, Champaign-Urbana, Illinois, May 2000), Bruce Berndt, et.al., editors, A.K. Peters Ltd, 2002. (pdf)

F. Luca and P.G. Walsh, Squares in Lehmer sequences with Diophantine applications, Acta Arithmetica 100 (2001), 47-62. (pdf)

F. Luca and P.G. Walsh, A generalization of a theorem of Cohn on the equations $x^3-Ny^2= \pm 1$, Rocky Mountain Journal of Mathematics 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, Mathematics of Computation 69 (2000), 1167-1182. (pdf)

P.G. Walsh, Irreducibility testing over local fields, Mathematics of Computation 69 (2000), 1183-1193. (pdf)

M.A. Bennett and P.G. Walsh, Simultaneous Pell equations with few or no solutions, Indagationes Mathematicae 11 (2000), 1-12. (pdf)

P.G. Walsh, Diophantine equations of the form $aX^4-bY^2= \pm 1$, in Algebraic Number Theory and Diophantine Analysis. (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. (pdf)

P.G. Walsh, On Diophantine equations of the form $(x^m-1)(y^n-1)=z^2$, in Tatra Mt. Math. Publ. 20 (2000), 1-3. (pdf)

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 72 (1999), 1-7. (pdf)

P.G. Walsh, The Diophantine equation $X^2-db^2Y^4=1$, Acta Arithmetica 57 (1999), 179-188. (pdf)

P. Ribenboim and P.G. Walsh, The ABC conjecture and the Powerful part of terms in binary recurring sequences. Journal of Number Theory 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 $b^2X^4-dY^2=1$. Proceedings of the AMS 127 (1999). (pdf)

P.G. Walsh, On the complexity of rational Puiseux expansions. Pacific Journal of Mathematics 188 (1999), 369-387.

A.J. van der Poorten and P.G. Walsh, A note on Jacobi symbols and continued fractions, American Mathematical Monthly 106 no. 1 (1999), 52-56.

P.G. Walsh, On a conjecture of Schinzel and Tijdeman. in Number Theory in Progress 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. SIAM Journal on Computing 28 (1999), 704-708.

P.G. Walsh, Two classes of simultaneous Pell equations with no solutions. Mathematics of Computation 68 (1999), 385-388.

P.G. Walsh, A note on Ljunggren's theorem about the Diophantine equation $aX^2-bY^4=1$. Comptes Rendues Mathematical Reports of the Royal Society of Canada 20 (1998), 113-119.

P.G. Walsh, On integer solutions to $x^2-dy^2=1,z^2-2dy^2=1$. Acta Arithmetica 87 (1997), 69-76.

P.G.Walsh, A quantitative version of Runge's theorem on Diophantine equations. Acta Arithmetica 62 (1992), 157-172.

Summer Undergraduate Research Papers

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. Acta Arithmetica 48 (1987), 341-345.

R.A.Mollin and P.G.Walsh, On non-square powerful numbers. Fibonacci Quarterly 25 (1987), 34-37.

R.A.Mollin and P.G.Walsh, On powerful numbers. International J. Math. and Math. Sci. {\bf 9} (1986), 801-806.

R.A.Mollin and P.G.Walsh, A note on quadratic fields, powerful numbers, and the Pellian. Comptes Rendues Mathematical Reports of the Royal Society of Canada 8 no. 2 (1986), 109-114.

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