## Chapter 1 : 4/n=1/a+1/b+1/c (Erdös - Strauss Conjecture : D11 Egyptian fractions)

(April 19, 1997) [Japanese]

### Abstract

1. Erdös and Strauss conjectured that the Diophantine equation
2. 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.

3. 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.

4. In order to understand the difficulty of this problem,
5. 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

1. How to solve this equation
2. Program
3. Table of solutions (computation results)
4. Parametrized solutions which contain the solution a=b or b=c
5. How to construct the parametrized solution whic contains the solution b=kp
6. Tables of the parametrized solutions of the case b=kp
7. The case that the solution could not find under a≤100
8. How to construct the parametrized solution from each solution (complete)
9. How to construct the parametrized solution from the solution b≠kp in heuristic way
10. Program
11. Table of solutions and parametrized solutions
12. Conclusion

E-mail : kc2h-msm@asahi-net.or.jp
Hisanori Mishima