— PROBLEM STATEMENT —
Thus is like the sum-of-divisors function , except it includes in its sum only the small divisors of . Whereas , the function is much smaller. In fact, the trivial bound gives
and the divisor bound for all gives . In particular, , so it makes sense to pose the following question:
Question: For which do we have ? Are there infinitely many for which this holds?
— SOME SIMPLE OBSERVATIONS —
The answer to the second half of our question is a resounding yes. The easiest way to prove this is to notice that for all primes , hence trivially.
So the real question we’d like to answer is the following:
Question: For which composite do we have ? Are there infinitely many composite for which this holds?
And now we have hit the nail on a non-trivial question. I suspect that the answer to the second half of this question is still yes, because (as we will see in a few sections), it follows from a laundry list of well-known conjectures in number theory.
But first, to get our feet wet, let’s prove some basic observations.
Proposition: If and is a prime power, then is prime.
Proposition: If and is a semiprime, then .
Proposition: If and is a product of three primes, then is divisible by a square and satisfies one of the following:
- (a) or
- (b) with prime,
- (c) with prime.
The first two Propositions are simple exercises and the third follows by casework on the size of , , and relative to . This third Proposition is particularly interesting because we see in cases (b) and (c) the potential for infinitely many composite solutions. In fact, the set would be infinite provided that the following conjecture holds:
Conjecture: There are infinitely many primes such that either or is simultaneously prime.
This conjecture is probably true (but far from being proven), in that it is a consequence of Schinzel’s Hypothesis H. Roughly, this conjecture implies that any `nice’ collection of distinct irreducible polynomials will simultaneously produce primes for infinitely many inputs.
Schinzel’s Hypothesis H is extended by the Bateman–Horn Conjecture, which predicts asymptotics for the number of inputs that produce these simultaneous primes. For each of the polynomials , this amounts to
choices of that make both and both prime. Of course, modern number theory is a far ways from proving either of these very general conjectures.
— OTHER PARAMETRIZED FAMILIES —
A natural next step could be the classification of all with four prime factors such that . I am wary of doing this for two reasons:
- The number of cases to consider as regards distinct or non-distinct primes grows to 5, the number of partitions of 4.
- Parametrized families for which many prime factors are made large in the limit will require Schinzel’s Hypothesis H again, or (at minimum) the Hardy–Littlewood k-tuple Conjecture.
For these reasons (which may be completely off-base), I think that a better route is to consider parametrized families of solutions in which only one or two prime factors is made large.
The simplest such family is one of the form , in which is some constant and is assumed to take on infinitely many prime values. For sufficiently large primes , we have
so we are led to consider such that . Yet in general, hence this forces , a contradiction.
The second simplest family would be of the form , in which and are expected to satisfy Schinzel’s Hypothesis H. But this will likely lead to another intractable problem unless we assume that and are both linear.
It’s hard to make sense of the outcome unless and have the same leading coefficient, so we suppose that and with . It turns out that this does lead to an infinite family (for certain ), which we describe below:
Theorem A: Choose such that . For any (sufficiently large) prime such that is also prime, we have for
Theorem A is best understood in terms of prime gaps. If is fixed and there exist infinitely many pairs of primes that differ by , then we have found an infinite set of solutions to .
The most famous conjecture about prime gaps is the Twin Prime Conjecture, which claims that the prime gap two occurs infinitely often. There’s no real connection between Theorem A and the Twin Prime Conjecture (since cannot be 2), but choosing gives and relates our problem to the infinitude of cousin primes, primes such that is also prime.
Corollary: If is a cousin prime, then satisfies .
Similarly, relates our question to the infinitude of prime pairs which differ by . It is a natural question to ask which (even) integers appear in the image of . It is not all of them (see the exercises). A list of all the admissible and the corresponding values for are given below:
We seem to have produced a necessary and sufficient condition for generating even perfect numbers. The sufficiency is of course easy to check, and we obtain the following:
Corollary: Fix and choose such that is a Mersenne prime. For any (sufficiently large) prime such that is also prime, we have for
Another fun tidbit is revealed along the way. Since these satisfy , we have in particular that . In other words,
Corollary: Let be an even perfect number. Then .
Alas, the existence of infinitely many perfect numbers (or equivalently, of infinitely many Mersenne primes) is another open problem. (It is, however, widely believed.)
— CONCLUSIONS? —
Unfortunately, I cannot yet resolve the question of the infinitude of the set of (composite) such that . The best we’ve done is show that it seems extremely likely to be true, and that it follows from any of the following statements:
- Schinzel’s Hypothesis H
- the Hardy–Littlewood k-tuple Conjecture
- there exist infinitely many cousin primes (or any other class of prime pairs which differ by one more than a Mersenne prime)
- there exist infinitely many Mersenne primes (or equivalently, there exist infinitely many even perfect numbers)
Still, I think that a clever argument might solve our problem without appeal to any of the conjectures listed above. My cautious optimism stems from the fact that Theorem A involves not just families of primes with gap 4, or of gap 28, but of gaps which lie in a somewhat large (and conjecturally infinite) set.
If other parametrized families of solutions to come to light which reduce our question to the infinitude of twin primes, and then to the infinitude of cousin primes, etc., up to the infinitude of primes which differ by , then partial progress towards the twin prime conjecture would show that for infinitely many composite .
More likely, it may be possible to show that our problem follows from the infinitude of simultaneous primes in the larger constellations , for in a sufficiently large class of -tuples. A theorem of Maynard (Theorem 1.2 in Small Gaps Between Primes) implies that at least one constellation would actually give infinitely many simultaneous primes, so long as the set of allowed -tuples was sufficiently large.
— EXERCISES —
Exercise 1. Prove that for any .
Exercise 2. Prove that for any . Hint: Find an effective lower bound for to make this a finite computation.
Exercise 3. Find a value of for which and that is not a perfect number. How many such are there less than ?
Exercise 4. Prove that no parametrized family of the form
or the form
satisfies for all sufficiently large primes for which each factor is prime.
Exercise 5. Prove that , where is twice a perfect number.