This chapter is a collection of some results from other areas of Number Theory which are applied in this book. Section 1.1 deals with estimates for prime counting functions \(\pi (x),\vartheta (x)\) and factorials. In Sect. 1.2, we state a theorem of Baker (Theorem 1.2.1) giving lower bounds for a linear form in logarithms of rational numbers with rational coefficients and a result of Shorey (Theorem 1.2.2) giving sharper lower bounds. Further we state two results of Baker (Theorem 1.2.3 and Theorem 1.2.4 ) on superelliptic equations and rational approximations to algebraic numbers followed by two applications due to Evertse (Lemma 1.2.5) and Bennett (Theorem 1.2.6). Section 1.3 gives a result from the works of Roth and Halberstam on k-free integers (Lemma 1.3.1). In Sect. 1.4, we record some results of Erdős based on sieve methods. In Sect. 1.5, we show how Erdős used elementary arguments to give upper bounds for products of consecutive primes. The last Sect. 1.6 gives a summary of some basic ideas in the area of Modular forms and important theorems such as Modularity Theorem for Elliptic curves (Theorem 1.6.10) and Ribet’s level lowering theorem (Theorem 1.6.12). Further we illustrate their application to Generalized Fermat Equations (Lemmas 1.6.14, 1.6.15 and 1.6.16).