A Brief Study of Odd Integers Between Squares and Adjacent Oblongs

In April 2020, I proposed a conjecture regarding the distribution of prime numbers within specific intervals defined by square and oblong numbers. This conjecture was first discussed in the comments on sequences (https://oeis.org/A307508) and (https://oeis.org/A334163) on the Online Encyclopedia of Integer Sequences (OEIS). Sequence (https://oeis.org/A307508) identifies the primes located between a square number and its following oblong number, while sequence (https://oeis.org/A334163) lists the primes situated between an oblong number and its following square number.Building on this foundation, the present study proposes the following extensions of Legendre's Conjecture (inspired by discussions on https://www.mersenneforum.org/node/21664 and https://www.mersenneforum.org/node/21573):•There are at least 3 prime numbers between any two consecutive odd squares. Starting from 9 (the first odd composite square), there are at least 5 prime numbers between any two consecutive odd squares.•Including the prime odd square-minus-1 number 3, there are at least 5 prime numbers between any two consecutive odd square-minus-1 numbers.•Including the even prime number 2, there are at least 5 prime numbers between any two consecutive 0212-Oblong numbers.•There are at least 5 prime numbers between any two consecutive 0620-Oblong numbers.•Each of the four zones (0212-SO, 0212-OS, 0620-SO, 0620-OS) contains at least one prime number along the number line. Furthermore, starting from the odd square-minus-1 number 15, the 0620-SO zone contains at least two prime numbers along the number line.