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| Format: | Preprint |
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2018
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| Online-Zugang: | https://arxiv.org/abs/1809.03328 |
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| _version_ | 1866915637616443392 |
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| author | Saxena, Anupam |
| author_facet | Saxena, Anupam |
| contents | For $p$ and $q$ any two distinct Fermat or Mersenne primes, $m,n,r$ as positive integers and $μ= \pm 1$ satisfying any diophantine relation, $\mbox{(i)}\; 2^m + μ= p^nq^r, \mbox{(ii)} \; 2^mp^n + μ= q^r \mbox{ or } \mbox{(iii)} \; p^n + μq^r = 2^m$, it is shown that the number of triplets $\{A, B, C \}$ with $\gcd(A,B) = 1$ and $C = A + B$, for which their product is of the form $ABC = 2^mp^nq^r$ and which satisfy $C > \mathrm{rad}(ABC)^{1 + \varepsilon}$ for any real $\varepsilon > 0$, is finite.
For the triplet $\{2^{y+1}, 2^{2y}+1, (2^y+1)^2\}$, a solution to (iii) with positive integer $y$ such that $2^y+1$ and $2^{2y}+1$ are primes, $\mathrm{rad}(ABC)^{1 + \varepsilon} > C$ holds for any $\varepsilon > 0$. Furthermore, finiteness of the number of solutions of (iii) when $n$ is even, is demonstrated elsewhere (Ref. [64]). All other solutions are enumerated. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_1809_03328 |
| institution | arXiv |
| publishDate | 2018 |
| record_format | arxiv |
| spellingShingle | ABC Conjecture: $ABC = 2^m p^n q^r$ with Fermat or Mersenne Primes Saxena, Anupam Number Theory For $p$ and $q$ any two distinct Fermat or Mersenne primes, $m,n,r$ as positive integers and $μ= \pm 1$ satisfying any diophantine relation, $\mbox{(i)}\; 2^m + μ= p^nq^r, \mbox{(ii)} \; 2^mp^n + μ= q^r \mbox{ or } \mbox{(iii)} \; p^n + μq^r = 2^m$, it is shown that the number of triplets $\{A, B, C \}$ with $\gcd(A,B) = 1$ and $C = A + B$, for which their product is of the form $ABC = 2^mp^nq^r$ and which satisfy $C > \mathrm{rad}(ABC)^{1 + \varepsilon}$ for any real $\varepsilon > 0$, is finite. For the triplet $\{2^{y+1}, 2^{2y}+1, (2^y+1)^2\}$, a solution to (iii) with positive integer $y$ such that $2^y+1$ and $2^{2y}+1$ are primes, $\mathrm{rad}(ABC)^{1 + \varepsilon} > C$ holds for any $\varepsilon > 0$. Furthermore, finiteness of the number of solutions of (iii) when $n$ is even, is demonstrated elsewhere (Ref. [64]). All other solutions are enumerated. |
| title | ABC Conjecture: $ABC = 2^m p^n q^r$ with Fermat or Mersenne Primes |
| topic | Number Theory |
| url | https://arxiv.org/abs/1809.03328 |