s998.py

from research import *
s=begin(998,'Fixed-pivot stagewise congruence','What condition makes both stages of a two-Key affine route integral?',{'first':'a+(p1/q1)(x−a)','second':'b+(p2/q2)(y−b)','assumptions':'reduced fractions; integer x,a,b'},['C997','C872–C889'])
proof={'first_stage':'x=a+q1·t; y=a+p1·t','second_stage':'p1·t≡b−a (mod q2)','solvability':'gcd(p1,q2) divides b−a','period':'q1·q2/gcd(p1,q2)'}
finish(s,proof,'The source pivots enter an exact congruence: an integral route exists iff gcd(p1,q2) divides their difference. When it exists, the domain is one residue class.','Evaluate the complete fixed nine-pair diagnostic with its two source-appointed pivots.',{'first_stage_reduction':F(25,23).denominator==23,'potential_obstruction':__import__('math').gcd(J.numerator,P.denominator)==3})