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Consecutive primes in tuples

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

Acta Arithmetica

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 (resp....

Consecutive square-free values of the type x 2 + y 2 + z 2 + k , x 2 + y 2 + z 2 + k + 1

Ya-Fang Feng (2023)

Czechoslovak Mathematical Journal

We show that for any given integer k there exist infinitely many consecutive square-free numbers of the type x 2 + y 2 + z 2 + k , x 2 + y 2 + z 2 + k + 1 . We also establish an asymptotic formula for 1 ≤ x , y , z ≤ H such that x 2 + y 2 + z 2 + k , x 2 + y 2 + z 2 + k + 1 are square-free. The method we used in this paper is due to Tolev.

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