MAIN RESULTS & NON-PRIMITIVE SOLUTIONS OF A GENERALIZED FERMAT EQUATION xm+yn = zr

ABSTRACT

The proposition was first conjectured by Pierre de Fermat in 1637 in the margin of a copy of Arithmetica; Fermat added that he had a proof that was too large to fit in the margin. However, there were doubts that he had a correct proof because his claim was published by his son without his consent and after his death. After 358 years of effort by mathematicians, the first successful proof was released in 1994 by Andrew Wiles, and formally published in 1995; it was described as a "stunning advance" in the citation for Wiles’s Abel Prize award in 2016. It also proved much of the modularity theorem and opened up entire new approaches to numerous other problems and mathematically powerful modularity lifting techniques. The unsolved problem stimulated the development of algebraic number theory in the 19th century and the proof of the modularity theorem in the 20th century. It is among the most notable theorems in the history of mathematics and prior to its proof was in the Guinness Book of World Records as the "most difficult mathematical problem" in part because the theorem has the largest number of unsuccessful proofs.  The equation x^m+yⁿ = z^r is considered under the condition that the given integers values for m, n, and r are greater than one. Solutions to this equation are given for cases in which gcd(mn; r) = 1, gcd(mr, n) = 1, or gcd(nr, m) = 1. 

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