Small gaps between three almost primes and almost prime powers
Tom 203 / 2022
Streszczenie
A positive integer is called an -number if it is the product of j distinct primes. We prove that there are infinitely many triples of E_2-numbers within a gap size of 32 and infinitely many triples of E_3-numbers within a gap size of 15. Assuming the Elliott–Halberstam conjecture for primes and E_2-numbers, we can improve these gaps to 12 and 5, respectively. We can obtain even smaller gaps for almost primes, almost prime powers, or integers having the same exponent pattern in the their prime factorizations. In particular, if d(x) denotes the number of divisors of x, we prove that there are integers a,b with 1\leq a \lt b \leq 9 such that d(x)=d(x+a)=d(x+b) = 192 for infinitely many x. Assuming Elliott–Halberstam, we prove that there are integers a,b with 1\leq a \lt b\leq 4 such that d(x)=d(x+a)=d(x+b)=24 for infinitely many x.