On the equation ,

pseudoperfect numbers,

and perfectly weighted graphs

Authors:
William Butske, Lynda M. Jaje and Daniel R. Mayernik

Journal:
Math. Comp. **69** (2000), 407-420

MSC (1991):
Primary 11D68; Secondary 11Y50, 05C50

Published electronically:
August 19, 1999

MathSciNet review:
1648363

Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: We present all solutions to the equation with at most eight primes, improve the bound on the nonsolvability of the Erdös-Moser equation , and discuss the computational search techniques used to generate examples of perfectly weighted graphs.

**[1]**D. Borwein, J. M. Borwein, P. B. Borwein, and R. Girgensohn,*Giuga’s conjecture on primality*, Amer. Math. Monthly**103**(1996), no. 1, 40–50. MR**1369150**, 10.2307/2975213**[2]**Lawrence Brenton and Robert R. Bruner,*On recursive solutions of a unit fraction equation*, J. Austral. Math. Soc. Ser. A**57**(1994), no. 3, 341–356. MR**1297008****[3]**Lawrence Brenton and Daniel Drucker,*On the number of solutions of ∑^{𝑠}ⱼ₌₁(1/𝑥ⱼ)+1/(𝑥₁\cdots𝑥_{𝑠})=1*, J. Number Theory**44**(1993), no. 1, 25–29. MR**1219482**, 10.1006/jnth.1993.1030**[4]**Lawrence Brenton and Daniel Drucker,*Perfect graphs and complex surface singularities with perfect local fundamental group*, Tohoku Math. J. (2)**41**(1989), no. 4, 507–525. MR**1025319**, 10.2748/tmj/1178227724**[5]**Lawrence Brenton and Richard Hill,*On the Diophantine equation 1=∑1/𝑛ᵢ+1/∏𝑛ᵢ and a class of homologically trivial complex surface singularities*, Pacific J. Math.**133**(1988), no. 1, 41–67. MR**936356****[6]**Lawrence Brenton and Mi-Kyung Joo,*On the system of congruences ∏_{𝑗≠𝑖}𝑛ⱼ≡1\bmod𝑛ᵢ*, Fibonacci Quart.**33**(1995), no. 3, 258–267. MR**1337798****[7]**Zhen Fu Cao, Rui Liu, and Liang Rui Zhang,*On the equation ∑^{𝑠}ⱼ₌₁(1/𝑥ⱼ)+(1/(𝑥₁\cdots𝑥_{𝑠}))=1 and Znám’s problem*, J. Number Theory**27**(1987), no. 2, 206–211. MR**909837**, 10.1016/0022-314X(87)90062-X**[8]**Richard K. Guy,*Unsolved problems in number theory*, 2nd ed., Problem Books in Mathematics, Springer-Verlag, New York, 1994. Unsolved Problems in Intuitive Mathematics, I. MR**1299330****[9]**Z. Ke and Q. Sun,*On the representation of by unit fractions*, Sichuan Daxue Xuebao**1**(1964), 13-29.**[10]**Pieter Moree,*Diophantine equations of Erdős-Moser type*, Bull. Austral. Math. Soc.**53**(1996), no. 2, 281–292. MR**1381770**, 10.1017/S0004972700017007**[11]**Leo Moser,*On the diophantine equation 1ⁿ+2ⁿ+3ⁿ+\cdots+(𝑚-1)ⁿ=𝑚ⁿ.*, Scripta Math.**19**(1953), 84–88. MR**0054627****[12]**David Mumford,*The topology of normal singularities of an algebraic surface and a criterion for simplicity*, Inst. Hautes Études Sci. Publ. Math.**9**(1961), 5–22. MR**0153682**

Retrieve articles in *Mathematics of Computation of the American Mathematical Society*
with MSC (1991):
11D68,
11Y50,
05C50

Retrieve articles in all journals with MSC (1991): 11D68, 11Y50, 05C50

Additional Information

**William Butske**

Affiliation:
Department of Mathematics, Wayne State University, 1150 FAB, Detroit, Michigan 48202

Address at time of publication:
Department of Mathematics, Purdue University, West Lafayette, Indiana 47906

Email:
butske@math.purdue.edu

**Lynda M. Jaje**

Affiliation:
Department of Mathematics, Wayne State University, 1150 FAB, Detroit, Michigan 48202

Address at time of publication:
EDS Office Centre, Mailstop 2061, 300 E. Big Beaver Road, Troy, Michigan 48083

Email:
lynda.jaje@eds.com

**Daniel R. Mayernik**

Affiliation:
Department of Mathematics, Wayne State University, 1150 FAB, Detroit, Michigan 48202

Email:
mayernik@math.wayne.edu

DOI:
http://dx.doi.org/10.1090/S0025-5718-99-01088-1

Received by editor(s):
June 19, 1996

Received by editor(s) in revised form:
March 17, 1998

Published electronically:
August 19, 1999

Article copyright:
© Copyright 1999
American Mathematical Society