# Smooth solutions to the abc equation: the xyz Conjecture - Mathematics > Number Theory

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Abstract: This paper studies integer solutions to the ABC equation A+B+C=0 in whichnone of A, B, C has a large prime factor. Set HA,B, C= max|A|,|B|,|C| andset the smoothness SA, B, C to be the largest prime factor of ABC. Weconsider primitive solutions gcdA, B, C=1 having smoothness no larger thana fixed power p of log H. Assuming the abc Conjecture we show that there arefinitely many solutions if p<1. We discuss a conditional result, showing thatthe Generalized Riemann Hypothesis GRH implies there are infinitely manyprimitive solutions when p>8. We sketch some details of the proof of the latterresult.

Author: ** Jeffrey C. Lagarias, K. Soundararajan**

Source: https://arxiv.org/