Karan Singh Grewal

Odd perfect numbers: the largest prime factor exceeds 10^9 and the second largest exceeds 10^6

We prove that every odd perfect number is divisible by a prime exceeding 10^9, improving the bound 10^8 of Goto and Ohno. The proof follows the method of Hagis and Cohen, Jenkins, and Goto and Ohno. Its computational part is organised so that every intermediate claim is either an...

Odd perfect numbers have at least two prime factors exceeding 10^6: statement and verification record

Statement of a computational theorem with its verification record. Theorem: every odd perfect number has at least two distinct prime factors exceeding 10^6; in particular its second largest prime factor exceeds 10^6. The previous bound on the second largest prime factor, 10^4, is...

Odd perfect numbers have a prime factor exceeding 10^9

We prove that every odd perfect number is divisible by a prime exceeding 10^9, improving the bound 10^8 of Goto and Ohno. The proof follows the method of Hagis and Cohen, Jenkins, and Goto and Ohno. Its computational part is organised so that every intermediate claim is either an...

Odd covering systems with moduli square-free at the primes up to 19

Balister, Bollobás, Morris, Sahasrabudhe and Tiba proved that no finite collection of arithmetic progressions with distinct odd square-free moduli covers the integers, and observed that their proof uses square-freeness only at the primes p ≤ 73, asking to what extent 73 can be re...