Nechemia Burshtein
Notes on Number Theory and Discrete Mathematics, ISSN 1310-5132
Volume 16, 2010, Number 4, Pages 1—5
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Nechemia Burshtein
117 Arlozorov Street, Tel Aviv 62098, Israel
Abstract
A complete demonstration of solutions of the above Diophantine equation is given when p < q are primes and α, β are positive integers. Among the several examples exhibited, Example 3 provides a new solution containing eighty-five even numbers xi all of which are of the required form. Certain questions and modifications of the equation are also discussed.
Keywords
- Diophantine equations
- Egyptian fractions
AMS Classification
- 11D68
References
- N. Burshtein. On distinct unit fractions whose sum equals 1, Discrete Math. 300
(2005) 213–217. - N. Burshtein. Improving solutions of Σi=1k 1/Xi = 1 with restrictions as required by Barbeau respectively by Johnson, Discrete Math. 306 (2006) 1438–1439.
- N. Burshtein. The equation Σi=19 1/Xi = 1 in distinct odd integers has only the five
known solutions, Journal of Number Theory 127 (2007), 136–144. - N. Burshtein. All the solutions of the equation Σi=111 1/Xi = 1 in distinct integers f the form xi ∈ 3α5β7γ, Discrete Math. 308 (2008), 4286–4292.
- N. Burshtein. An improved solution of Σi=1k 1/Xi = 1 in distinct integers when xi ∤ xj for i ≠ j, Notes on Number Theory and Discrete Mathematics 16, (2010), 2, 1–4.
- C. Rivera, J. Ayala. The prime puzzles & problems connection – problem 35,
http://www.primepuzzles.net/.
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Cite this paper
Burshtein, N. (2010). On the Diophantine equation Σi=1k 1/Xi = 1 in distinct integers of the form xi pαqβ. Notes on Number Theory and Discrete Mathematics, 16(4), 1-5.