Proof of the ABC conjecture By Samuel Bonaya Buya Teacher Ngao girls’ secondary school Kenya Abstract In this research a short proof of the abc conjecture is presented. It is shown that the product of the distinct prime factors of ABC is greater than the square-root of c. An identity connecting c and rad(abc) is used to used to establish the lower limit value of rad(abc) in relationship to c. Key words Proof of the abc conjecture; Diophantine analysis Introduction The abc conjecture was first presented by David Masser (1985) and Joseph Oesterlé (1988). The conjecture is stated in terms of three positive integers a, b, c that are relatively prime that satisfy the a b c   . If d denotes the product of distinct prime factors of abc, the conjecture states that d is usually not much smaller than c. On 30 th August 2012 Shinichi Mochizuki released four preprints that developed inter-universal Teichmüller theory and use it to prove solve several problems in Diophantine geometry and in number theory. In the preprints Mochizuki proposed a proof of the abc conjecture. Six years along the line the proof has been understood by very few mathematicians. In this research a short and simple proof of the conjecture will be presented. Methodology Consider three coprime positive integers a, b and c such that; (1) a b c   The product of the radicals of the three integers is given by: () () () ( ) 2 d rad a rad b rad c rad abc  