DENSITY BOUNDS FOR THE 3X+1 PROBLEM .2. KRASIKOV INEQUALITIES

The 3x + 1 function T(x) takes the values (3x + 1)/2 if x is odd and x /2 if x is even. Let a be any integer with a not equal 0 (mod 3). If p i(a)(x) counts the number of n with \n\ less than or equal to x which eventually reach a under iteration by T, then for all sufficiently lar ge x, pi(a)(x) greater than or equal to x(.81). The proof is based on solving nonlinear programming problems constructed using difference in equalities of Krasikov.