Displaying similar documents to “Generalizing a theorem of Schur”

Consecutive primes in tuples

William D. Banks, Tristan Freiberg, Caroline L. Turnage-Butterbaugh (2015)

Acta Arithmetica

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In a stunning new advance towards the Prime k-Tuple Conjecture, Maynard and Tao have shown that if k is sufficiently large in terms of m, then for an admissible k-tuple ( x ) = g x + h j j = 1 k of linear forms in ℤ[x], the set ( n ) = g n + h j j = 1 k contains at least m primes for infinitely many n ∈ ℕ. In this note, we deduce that ( n ) = g n + h j j = 1 k contains at least m consecutive primes for infinitely many n ∈ ℕ. We answer an old question of Erdős and Turán by producing strings of m + 1 consecutive primes whose successive gaps δ 1 , . . . , δ m form an increasing...

On a divisibility problem

Shichun Yang, Florian Luca, Alain Togbé (2019)

Mathematica Bohemica

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Let p 1 , p 2 , ⋯ be the sequence of all primes in ascending order. Using explicit estimates from the prime number theory, we show that if k ≥ 5 , then ( p k + 1 - 1 ) ! ∣ ( 1 2 ( p k + 1 - 1 ) ) ! p k ! , which improves a previous result of the second author.

Truncatable primes and unavoidable sets of divisors

Artūras Dubickas (2006)

Acta Mathematica Universitatis Ostraviensis

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We are interested whether there is a nonnegative integer u 0 and an infinite sequence of digits u 1 , u 2 , u 3 , ⋯ in base b such that the numbers u 0 b n + u 1 b n - 1 + ⋯ + u n - 1 b + u n , where n = 0 , 1 , 2 , ⋯ , are all prime or at least do not have prime divisors in a finite set of prime numbers S . If any such sequence contains infinitely many elements divisible by at least one prime number p ∈ S , then we call the set S unavoidable with respect to b . It was proved earlier that unavoidable sets in base b exist if b ∈ { 2 , 3 , 4 , 6 } , and that no unavoidable set exists in base b = 5 . Now,...

Composite positive integers whose sum of prime factors is prime

Florian Luca, Damon Moodley (2020)

Archivum Mathematicum

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In this note, we show that the counting function of the number of composite positive integers n ≤ x such that β ( n ) = ∑ p ∣ n p is a prime is of order of magnitude at least x / ( log x ) 3 and at most x / log x .