Problems & Puzzles: Puzzles

 Puzzle 765. pa = qb + rc.sd Philippe Fondanaiche sent the following nice puzzle:   Let p, q, r and s be four distinct prime numbers chosen among the set {2,3,5,7} and we look for all the 4-uples (a, b, c, d) of integers > 0  satisfying the equation  (E) :  pa = qb + rc.sd   For example with p=7, q = 2, r = 3 and s = 5, we get a = 2, b = 2,c = 2 and d = 1 corresponding to the relation 7˛ =2˛ + 3˛.5 = 49   There are 4! = 24 possible ways to choose the four values of p, q, r and s which give 12 different equations (E).   Q1 Prove that among these 12 equations there are at least 10 of them for which there exists at least one solution in (a, b, c, d). żAre there equations with no solution?   Q2 For each equation having at least one solution, is there a finite or infinite number of solutions?  If there is a finite number of solutions, provide all the possible solutions.

Contributions came from Emmanuel Vantieghem

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Emmanuel wrote:

These are all the solutions I could find :

{p, q, r, s}      {a, b, c, d}

-----------------------------------
{2, 3, 5, 7}     {8, 4, 2, 1}
{2, 5, 3, 7}        -
{2, 7, 3, 5}     {6, 2, 1, 1}
{3, 2, 5, 7}       -
{3, 5, 2, 7}     {4, 2, 3, 1}
{3, 7, 2, 5}     {3, 1, 2, 1}
{5, 2, 3, 7}     {2, 2, 1, 1}
{5, 3, 2, 7}     {3, 3, 1, 2}
{5, 7, 2, 3}     {2, 1, 1, 2}
"              {4, 2, 6, 2}
{7, 2, 3, 5}     {2, 2, 2, 1}
{7, 3, 2, 5}     {2, 2, 3, 1}
"             {3, 5, 2, 2}
{7, 5, 2, 3}     {2, 2, 3, 1}

I think that there are no more.  If there is one it will be one with  a > 5000.

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