Problems & Puzzles: Puzzles

Puzzle 370. The first 13 numbers Muneer Jebreel Karama sent the following puzzle: He sent one solution with (p,q,r,s) = (5,3,2 11) Questions. Find more solutions.

---

Jan Van Delden wrote (June 2007):

I'm rather interested in the motivation underlying this problem, apart from the fact that one solution is easy to find. And apparently another quite hard. (Although I didn't use a number cruncher to be frank; I tested for p,q <50, very small for this site's standards I admit).

 

It seems to me that the restrictions don't allow for a lot of solutions. If we allow to loosen the restrictions it still will be difficult to find large solutions.

 

With: S_p[n]=Sum(i^p,i=1..n)

 

The problem states:

I)    S_p[13]=s^r S_q[13]

II)   S_p[13]=k^r

 

The first restriction will be hard enough as it is.

 

First point:

 

For the second restriction:

S_p[n] = k^2 seems to give a solution with p=3 for all n.

 

This is no surprise since S_3[n]=S_1[n]^2; Nichomachus's theorem.

 

Second point:

 

S_p[13] will be of the form 7^k * 13^l * remaining term, k and l>=1.

 

[If n prime then S_p[n] is congruent to sum(i,i=1..n) mod p, which gives S_p[13] congruent to 7*13 mod p, so at least 0 mod 7 and 0 mod 13]

 

Testing for several p shows the remaining term is not very likely to have the same factors, where the biggest one grows rather fast. Furthermore the the multiplicity of the factors 7 and 13 don't appear to have a 'regular' pattern (I missed it).

 

So we might be 'lucky' to find a second solution with r=2, let alone a different r>2 prime.

 

If we alter the problem in the form:

 

S_p[s] = r S_q[s] with p,q,r,s prime

 

It's not that hard to find small solutions for certain combinations of p,q;

For instance:

 

(5,3,37,7)

(7,3,1531,7)

(7,3,71947,19)

(5,3,661,31)

(7,3,1788571,43)

 

It will be hard enough to find large ones.

 

Even finding large solutions to t S_p[s] = r S_q[s] with p,q,r,s,t prime will require some crunching.

 

Then again, I'm not a number theorist, but as posed it looks a bit too much 'Wieferich' to me.

***

 

---