The well-known Bertrand’s postulate states that there is a prime between x and 2x for any real number \(x>1.\) A proof of this fact was later found by Chebyshev. Since then finding prime in short intervals has been the center problem in Prime Number Theory. This is a difficult problem. We may relax this problem and ask if a sequence of integers can have an integer which has a large prime divisor. To make the problem more tractable, one may restrict to consecutive integers or integers in an arithmetic progression. A positive assertion and proof was first given by Sylvester [1] in 1892. In his own words, his proof is wearisome. Schur [2] rediscovered and proved Sylvester’s result again in 1929. In the case of consecutive integers the proofs were simplified by Erdős [3] in 1934. Sylvester’s theorem for arithmetic progression was given a complete stature by Shorey and Tijdeman [4] in 1990. In this chapter, we shall state Sylvester’s theorem for consecutive integers and give the proof of Erdős (Theorem 3.1.1). For the case of arithmetic progression, we shall present the result of Shorey and Tijdeman (Theorem 3.1.2). The proof given here is self-contained in the sense that it does not assume previously known results for arithmetic progressions of small length. We apply Sylvester’s theorem to prove the irreducibility of truncated exponential polynomials (Theorem 3.4.1). Some extensions and refinements of Sylvester’s theorem are given in Sect. 3.5.