Legendre, Grimm, Balanced Prime, Prime triplet, Polignac's conjecture, a problem and 17 tips with proof to solve problems on number theory

In this paper we will prove five conjectures, one problem viz. 1) Legendre Conjecture. 2) Grimm's Conjecture. 3) Balanced Primes are infinite. 4) Prime triplets are infinite. 5) Polignac's conjecture. 6) Can a prime p satisfy 2p − 1 ≡ 1 (mod p 2 ) and 3p − 1 ≡ 1 (mod p 2 ) simultaneously? And 17 tips with proof to solve problems on number theory.