Abstract:
The following system of equations fx1 x1 = x2, x2 x2 = x3, 22x1 = x3, x4 x5 = x2, x6 x7 = x2g has exactly one solution in (N n f0; 1g)7, namely (2; 4; 16; 2; 2; 2; 2). Conjecture 2.1 states that if a system S of equations has at most five equations and at most finitely many solutions in (N n2fx0j; 1=g)x7,k t:hje;nk e2achf1s;u::c:h; 7sgoglu.tCioonnj(exc1t;u:r:e: ;2x.17)imsaptliisefisesthxa1t; t: h::e;rxe7a6re1i6n,fiwnhiteerlye S fxi xj = xk : i; j; k 2 f1; : : : ; 7g2gn[ f2 many composite numbers of the form 2 + 1. Conjectures 3.1 and 4.1 are of similar kind. Conjecture 3.1 implies that if the equation x! + 1 = y2 has at most finitely many solutions in positive integers x and y, then each such solution (x; y) belongs to the set f(4; 5); (5; 11); (7; 71)g. Conjecture 4.1 implies that if the equation x(x + 1) = y! has at most finitely many solutions in positive integers x and y, then each such solution (x; y) belongs to the set f(1; 2); (2; 3)g. Semi-algorithms semj (j = 2; 3; 4) that never terminate are described. For every j 2 f2; 3; 4g, if Conjecture j.1 is true, then semj endlessly prints consecutive positive integers starting from 1. For every j 2 f2; 3; 4g, if Conjecture j.1 is false, then semj prints a finite number (including zero) of consecutive positive integers starting from 1.
Keywords:
Erdo equation xx + 1 = y
,
composite Fermat numbers
,
Brocards problem
,
BrocardRamanujan equation