On Some of My Conjectures in Number Theory and Combinatorics

During my very long life I have made many conjectures in these subjects and have written several papers with similar titles. To avoid repetition as much as possible I will mainly mention problems where significant progress has been made in the last year. I start with an old conjecture of mine on the divisors of numbers. 1 .Denote by T(n) the number of divisors of n. 1 = d 1 < d 2 <. .. <d T (n) = n i n. I conjectured about 45 years ago that for almost all n (i .e .,for all n, neglecting a sequence of density 1)) we have (1) min d i+1/ d i My first idea was to attack (1) as follows : Let m • e primitive with respect to the property (1) i£ m satisfies (1) but no proper divisor of m satisfies it. Let u I <u 2 <. . be the sequence of primitive numbers. Clearly the integer^ satisfying (1) are the multiples of the u's. Thus to prove my conjecture it would only be necessary to prove that the set of multiples of the u's has a density and that this density is 1. This method was successful for the primitive abundant numbers, but the sum of the reciprocals of the primitive abundant numbers converges and the density of the abundant numbers is <1 [1j. Here I could p ro-.r e that L Land that the den-u .