Chapter 1 : 4/n=1/a+1/b+1/c
(Erdös - Strauss Conjecture : D11 Egyptian fractions)
(April 19, 1997) [Japanese]
Abstract
- Erdös and Strauss conjectured that the Diophantine equation
4/n=1/a+1/b+1/c
could be solved in positive integers for all n < 1. (still not be solved.)
It is checked n≤109.
- In this chapter, I will show,
- when b=kn, we can construct a parametrized solution
by the solution n, a, b, c.
- when b≠kn, we can construct a parametrized solution
by the solution n, a, b, c in heuristic way.
for the denominator is 4, 5, 6, 7 and any integer.
Applying this method, we can extend the search range dramatically.
- In order to understand the difficulty of this problem,
please try to search the solutions of following Diophantine equations.
- Q1 : 4/11=1/a+1/b+1/c
- Q2 : 5/11=1/a+1/b+1/c
- Q3 : 6/11=1/a+1/b+1/c
- Q4 : 7/11=1/a+1/b+1/c
- Q5 : 8/11=1/a+1/b+1/c
Contents
- How to solve this equation
- Program
- Table of solutions (computation results)
- Parametrized solutions which contain the solution a=b or b=c
- How to construct the parametrized solution whic contains the solution b=kp
- Tables of the parametrized solutions of the case b=kp
- The case that the solution could not find under a≤100
- How to construct the parametrized solution from each solution (complete)
- How to construct the parametrized solution from the solution b≠kp in heuristic way
- Program
- Table of solutions and parametrized solutions
- Conclusion
E-mail : kc2h-msm@asahi-net.or.jpHisanori Mishima