IPM Positions

Non Resident Researcher (non-resident), School of Mathematics (2010 - 2012 )

Past IPM Positions

Associate Researcher (non-resident), School of Mathematics (2006 - 2009) Associate Researcher (non-resident), School of Mathematics (2004 - 2006) Associate Researcher (non-resident), School of Mathematics (2002 - 2003)

Non IPM Affiliations

Associate Professor of Shahrekord University

Research Interests

Number Theory in general, Diophantine Equations, Diophantine Approximations.

Research Activities

Problem 1. It is well known that every rational number is a sum of three rational cubes. Thus it is an interesting question to determine the rational numbers which are sums of two rational cubes, or if we prefer, the integers A for which the equation ? x3+y3=Az3 (1) is solvable in integers. This equation has attracted the attention of many mathematicians over a period of years. Euler was first who proved that the equation (1) is impossible if a=1 and a=4, and that x=?y if a=2. Fundamental contribution to the investigation of such kind equations was made by Sylvester, van der Corput, Delone and Faddeev. Author proved that the equation ? x3+y3=pqz3 has no solutions in nonzero integers x, y, z if ?p, q are primes of special kind. Note that a very interesting question for this problem is a search of other values of a for which the equation (1) has no solutions in nonzero integers x, y, z. Problem 2. To investigate Diophantine equation ? xp-yq=k, where the unknowns x,?y,?p and q are integers all ? 2, in some particular cases. In general case conjecture Pillai says that this equation has only finitely many solutions (x, y, p, q). This means that in the increasing sequence of perfect powers xp, with x ? 2 and p ? 2 the difference between two consecutive terms tends to infinity. It is not even known that for, say, k=2, Pillai''s equation has only finitely many solutions. A related open question is whether the number 6 occurs as a difference between two perfect powers: Is there a solution to the Diophantine equation xp-yq=6?

Present Research Project at IPM

Diophantine equation

Related Papers

1.	Kh. Hessami Pilehrood and Kh. Hessami Pilehrood Bivariate identities for values of the Hurwitz zeta function and supercongruences Electron. J. Combin. 18 (2012), #P35  [abstract]

2.	Kh. Hessami Pilehrood and T. Hessami Pilehrood On a continued fraction expansion for Euler's constant J. Number Theory  (Accepted) [abstract]

3.	Kh. Hessami Pilehrood and T. Hessami Pilehrood Rational approximations for values of Bell polynomials at points involving Euler s constant and zeta values Austral. Math. Soc. (Accepted) [abstract]

4.	T. Hessami Pilehrood and Kh. Hessami Pilehrood Rational approximations for values of the digamma function and a denominators conjecture Math. Notes (Accepted) [abstract]

5.	Kh. Hessami Pilehrood and T. Hessami Pilehrood A q-analogue of the bailey-Borwein-Bradley identity Journal of Symbolic Computation 46 (2011), 699-711  [abstract]

6.	Kh. Hessami Pilehrood and T. Hessami Pilehrood Vacca-type series for values of the generalized Euler constant function and its derivative Journal of Integer Sequences 13 (2010), 10.7.3  [abstract]

7.	Kh. Hessami Pilehrood and T. Hessami Pilehrood Rational approximations for the quotient of gamma values Indag. Math. (N.S.) 20 (2009), 583-601  [abstract]

8.	Kh. Hessami Pilehrood and T. Hessami Pilehrood An Apery-like continued fraction for πcothπ J. Difference Equ. Appl. (Accepted) [abstract]

9.	Kh. Hessami Pilehrood and T. Hessamo Pilehrood Generating function identities for ζ(2n+2), zeta(2n+3) via the WZ method Electron. J. Combin. 15 (2008), 1-9  [abstract]

10.	T. Hessami Pilehrood and Kh. Hessami Pilehrood On a conjecture of Erdos Math. Notes 83 (2008), 281-284  [abstract]

11.	T. Hessami Pilehrood and Kh. Hessami Pilehrood Infinite sums as linear combinations of polygamma functions Acta Arith. 130 (2007), 231-254  [abstract]

12.	Kh. Hessami Pilehrood and T. Hessami Pilehrood Apéry-like recursion and continued fraction for π coth π ( In: Diophantische approximationen) [abstract]

13.	Kh. Hessami Pilehrood and T. Hessami Pilehrood On conditional irrationality measures for values of the digamma fuction J. Number Theory 123 (2007), 241-253  [abstract]

14.	Kh. Hessami Pilehrood and T. Hessami Pilehrood Arithmetical properties of some series with logarithmic coefficient Math. Z. 255 (2007), 117-131  [abstract]

15.	T. Hessami Pilehrood and Kh. Hessami Pilehrood Irrationality of the sums of zeta values Math. Notes 79 (2006), 607-618  [abstract]

16.	Kh. Hessami Pilehrood and T. Hessami Pilehrood On sums of two rational cubes Indian J. Pure Appl. Math. 36 (2006), 707-717  [abstract]

17.	Kh. Hessami Pilehrood and T. Hessami Pilehrood (Joint with W. Zudilin) Irrationality of certain numbers that contain values of the di- and trilogarithm Math. Z. 254 (2006), 299-313  [abstract]

18.	Kh. Hessami Pilehrood and T. Hessami Pilehrood On the diophantine equation x2+3=pyn Indian J. Pure Appl. Math. 36 (2005), 431-439  [abstract]

19.	T. Hessami Pilehrood and Kh. Hessami Pilehrood Lower Bounds for linear forms in values of polylogarithms Math. Notes 77 (2005), 573-579  [abstract]

20.	T. Hessami Pilehrood and Kh. Hessami Pilehrood Criteria for irrationality of generalized Euler's constant J. Number Theory 108 (2004), 169-185  [abstract]