Salacious numbers

Numbers divisible by the product of their prime factors.

Salacious numbers are rare and fascinating things indeed. Although there exist many superficially similar classes of numbers, such as 'perfect' numbers (equal to the sum of their prime factors), 'friendly' numbers (pairs of numbers equal to the sum of each other's prime factors), 'greedy' numbers (equal to the sum of all prime factors except their own), 'comely' numbers (equal to the product of all of their prime factors that look good in base 23), etc., these have no real mathematical significance and are really just numerological fancies.

Salacious numbers however, are different. They have relevance to several areas of modern research, and have many tantalising properties that make them rewarding, if elusive, beasts to study. For example, the following things are known about them:

• There are infinitely many salacious numbers.

• For any n, there are infinitely many salacious numbers that are multiples of n.

• The proportion of salacious numbers less than n remains roughly constant as n grows (this is Stobbs's Salacious Distribution Theorem).

But lest you think that these are well-understood little integers, the following hypotheses should snap you out of your complacent day-dreaming. None of these are proven, but all have extensive evidence in their favour and few (if any) counter-examples:

• There is at least one salacious number between any two non-consecutive primes.

• Every even number can be written as the sum of two salacious numbers (Gubbon's Conjecture)

• For all n there is a solution to xⁿ + yⁿ = zⁿ where x,y,z are salacious numbers.

An intruiging topic then, but one containing many traps for the casual dummlichter!.