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On the greatest prime factor of \(2^p-1\) for a prime \(p\) and other expressions. (English) Zbl 0296.10021
It is proved that for almost all primes \(p\), the greatest prime factor of \((2^p-1)\) exceeds \(cp \left({\log p \over \log \log p}\right)^2\) where \(c>0\) is an absolute constant. The proof depends on Brun’s sieve method and Baker’s theory on linear forms in the logarithms of algebraic numbers. Some results relating to the greatest prime factor of other arithmetic expressions are also proved.
Reviewer: Paul Erdős

MSC:
11A51 Factorization; primality
11N36 Applications of sieve methods
11J86 Linear forms in logarithms; Baker’s method
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