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1. For a natural number a, denote by P(a) the greatest prime factor of a. Stewart [10] proved that there exists an effectively computabl e constant c > 0 such that P(2"—1) (logp) 1 ~4 p for all primes p > c. In § 2, we shall prove that P(2'-1) /p exceeds constant times log p for all primes. In § 5, we shall prove that foràlmost all' primes p , P(2"-1) (logp) 2 (2) p (loglogp) 3 For the definition of 'almost all', see § 5. Let u > 3 and k > 2 be integer s and denote by P(u, Ic) the greatest prime factor of (u+1). .. (u + k). It follows from Mahler's work [6a] that P(u, 1c) > loglogu. See also [6 ] and [8]. In § 4, we shall show that for u > 1c 312 P(u, 1c) > c 1 kloglog u where cl > 0 is a constant independent of u and k. It follows from well-known results on differences between consecutive primes that P(u, k) u +1 whenever k u 7c 3I2. Let a < b be positive integers which ar e composed of the same primes. Then, in § 3, we shall show that there exis t positive constants c 2 and c3 such tha t b—a> c,(log a)'3. Erdös and Selfridge [5] conjectured that there exists a prime between a and b. The proof of all these theorems depend on the following recent resul t on linear forms in the logarithms of algebraic numbers. Let n > 1 be an integer. Let a l ,. . ., an be non-zero algebraic number s of heights less than or equal to A 1 ,. . ., A n respectively, where each Az > 27 .

On the greatest prime factor of $2^p - 1$ for a prime p and other expressions