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The Uniqueness of the Prime Markoff Numbers
Author(s) -
Button J. O.
Publication year - 1998
Publication title -
journal of the london mathematical society
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 1.441
H-Index - 62
eISSN - 1469-7750
pISSN - 0024-6107
DOI - 10.1112/s0024610798006292
Subject(s) - mathematics , number theory , conjecture , prime number , uniqueness , element (criminal law) , diophantine equation , prime (order theory) , algebraic number , natural number , order (exchange) , prime number theorem , discrete mathematics , combinatorics , pure mathematics , mathematical analysis , law , finance , political science , economics
Given the Diophantine equation a 2 + b 2 + c 2 =3 abc , a solution triple of natural numbers ( a , b , c ) can be arranged in ascending order so that a ⩽ b ⩽ c . Then, given the largest element c , one can ask whether this uniquely determines the triple. This is referred to as the Markoff conjecture. The paper proves that, if c is prime, then there is indeed only one triple that solves the equation with c as the largest element. The proof uses only standard algebraic number theory, but it was prompted by geometric considerations.