Giuga's conjecture (a type of converse) .

This is a well-known, longstanding problem (one on which I spent a lot of time over the years). I first came upon it at school, in Sierpinski's wonderful 1964 collection of problems. There, on page 111, Sierpinski asks:

Does there exist a composite number n which is a divisor of the number

+ ... +

and Sierpinski goes on to (trivially) remark: It is easy to prove that if p is a prime number, then

+ ... +

is divisible by p . [and that is all Sierpinski had to say about it.]

The point of Sierpinski's remark is that, for prime p , one has:

(mod p ), (mod p), ... , (mod p)

and so, trivially: + ... + (mod p )

Thus the thrust of Sierpinski's question is: does

+ ... + (mod n ) ... (c)

hold only for prime n ?

Comment . It is well known (as indeed anyone will establish who tackles this question) that a composite n satisfying (c) must have these properties:

n is odd and squarefree (so , for distinct odd primes , ... , , )

divides for every , ... , r

Now, of course, one would like to show the latter is impossible...

A comment . An immediate consequence of the latter is that:

+ ... + is an integer ... (e)

One might be tempted to argue that is impossible... , but one should be aware that it will not be easy . One cannot argue it is impossible merely for distinct primes, for one should be aware that:

So, is (e) impossible for distinct odd primes?

Recommended reading . The January 1996 American Mathematical Monthly article by Borwein (D., J. M., and R.) and Girgensohn, R., who attribute the above question to G. Guiga (1950), where they list ten associated Open Problems.