Paper 12. Paired Almost Primes in Square Windows and in Short Intervals

Abstract. Every sufficiently large window between consecutive squares carries pairs (a, a+2) with at most four prime factors on each side, and so does every interval of length x^0.499; this comes from an overlap weight that gives back part of the penalty when both members carry middle prime factors. The plain Richert weight gives at most five on each side in the square window, at most seven in total with a different constant, and three on each side in intervals of length x^0.78. Every coefficient is certified by interval arithmetic. This part uses no object from the cell system.