HAL Id: hal-02181628
https://hal.archives-ouvertes.fr/hal-02181628
Preprint submitted on 12 Jul 2019
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Proof of the abc-Conjecture
Constantin M. Petridi
To cite this version:
Constantin M. Petridi. Proof of the abc-Conjecture. 2019. �hal-02181628�
Proof of the abc-Conjecture
Constantin M. Petridi cpetridi@math.uoa.gr
Abstract
We prove the abc-Conjecture. The proof is based on our paper [3] and our paper [4].
1 Proof of the abc-Conjecture
We consider the Diophantine equation
a1x1+a2x2 +· · ·+aNxN =n (1)
where
ai := positive integers, i= 1, 2,· · · , N , N := a positive integer, greater than one,
n:= a positive integer greater than one.
Before starting the proof and in order to facilitate the reader we quote Theorem 4 of our paper [4].
“Theorem 4. For any positive integer c there is an integer N =N(c), 1 ≤ N < φ(c)
2 , such that at least N of the radicals, figuring in (1), denoted generally by R(abc) satisfy
R(abc) > kc,
where k the absolute constant of Theorem 3.”
1
We now revert to the abc-Conjecture. We consider the Diophantine equation
a1x1+a2x2 +· · ·+aNxN =n (2)
where
ai := positive integers, i= 1, 2,· · · , N , N := a positive integer, greater than one,
n:= a positive integer greater than one.
As all xi, i = 0, 1,· · · , N can not all be zero, since n is a positive integer, greater than one, we conclude that there is at least one which is not zero. But this is precisely the one mentioned in the quotation.
References
[1] http://en.wikipedia.org/wiki/Abcconjecture
[2] Principle of Inclusion and Exclusion Principle (PIE) — Brilliant Math and Sci- ence Wiki
[3] Constantin M. Petridi, The sum of thek-th power Euler set and the connection with Artin’s conjecture for primitive roots [arXiv: 1612016632 v2 [math. NT] 26 February 2018].
[4] Constantin M. Petridi, The number of equations c = a+b satisfying the abc- Conjecture [arXiv: 0904.1935 v1 [MATH.NT] 13 Apr 2009].
[5] Constantin m. Petridi, A strong “abc-Conjecture” for certain partitionsa+b of c[arXiv: MATH/0511224 v1 [MATH.NT] 9 Nov 2005].
[6] https://en.wikipedia.org/wiki/Chebychev%27sinequality.
