250 Problems in Elementary Number Theory - Sierpinski (1970)

PROBLEMS 15167*. Prove that for every positive integer m, for all sufficiently large s,the equation1 1 1...m+m+ ... +...m = 1-"I X2 -"shas at least one solution in positive integers Xl> Xz, •.• , x •.168. Prove that for every positive integer s the equation1 1 1 1_..2 + _.2 + ... + _.2 = TAi X2 ~.+1has infinitely many solutions in positive integers Xl> X2, ••• , Xu X.H'169. Prove that for every integer s ~ 3 the equationhas infinitely many solutions in positive integers Xl> X2, ... , Xu X.+I'170* . Find all integer solutions of the system of equations171. . Investigate, by elementary means, for which positive integers n theequation 3x+5y = n has at least one solution X, y in positive integers, andprove that the number of such solutions increases to infinity with n.172. Find all solutions in positive integers n, X, y, z of the equationnX+nY = n"'.173. Prove that for every system of positive integers m, n there existsa linear equation ax+by = c, where a, b, c are integers, such that the onlysolution in positive integers of this equation is X = n, y = m.174. Prove that for every positive integer m there exists a linear equationax+by = c (with integer a, b, and c) which has exactly m solutions in positiveintegers x, y.175. Prove that the equation r+y2+2xy-mx-my-m-1 = 0, wherem is a given positive integer, has exactly m solutions in positive integers x, y.