Monthly Archives: August 2016

When {(x+y+z)^5-x, (x+y+z)^5-y, (x+y+z)^5-z} are squares

    Find integers     such that ,      ,        are squares, where                                                 … Continue reading

Posted in Number Puzzles | Tagged | Leave a comment

Num3er 666 and equal sums and sum of 5-th powers

        Note that                                              

Posted in Number Puzzles | Tagged | Leave a comment

Three numbers in A.P. whose sum is a 6-th power

      Find three numbers in arithmetical progression whose sum is a 6-th power.                                            

Posted in Number Puzzles | Tagged | 1 Comment

1 + x + … + x^m = 1 + y + … + y^n

    It appears that for   holds only for in addition to evident solutions if     or     is negative.                                   … Continue reading

Posted in Number Puzzles | Tagged | Leave a comment

Pell identities: (P(n))^4+(P(n+1))^4+(P(n+2))^4 = 2((P(n))^2+2P(n)P(n+1)+4(P(n+1))^2)^2

  Prove the following identities:       Let     and     be any real numbers Then   can be confirmed algebraically. This identity has an interesting application to the Pell family. Let     be an integer … Continue reading

Posted in Number Puzzles | Tagged | 2 Comments

Powers of 2 and 3 as a sum of two powers

                  and so on.   and so on.   and so on.                                    

Posted in Number Puzzles | Tagged , | Leave a comment

Primitive Pythagorean triples (a, b=a+1, c)

    Let P be the perimeter   (1)   Can you explain the following: The generators     and       of the first few triangles are the Pell numbers 2,   5,   12,   29, …, … Continue reading

Posted in Number Puzzles | Tagged | 2 Comments

Make {x^2 + y^2 + z^2, x^2 y^2 + x^2 z^2 + y^2 z^2} squares – Part 2

    Find three positive integers     to can make the two expressions squares       Here’s a parametric solution established by Euler:   In two ways:                         … Continue reading

Posted in Number Puzzles | Tagged | Leave a comment

Make {x^2 + y^2 + z^2, x^2 y^2 + x^2 z^2 + y^2 z^2} squares — Part 1

    Find three positive integers     to can make the two expressions squares                                                                                              Here are some solutions:                                   … Continue reading

Posted in Number Puzzles | Tagged | 2 Comments

Make {2x^2 + y^2 + z^2, x^2 + 2y^2 + z^2, x^2 + y^2 + 2z^2} squares

    Find three positive integers     to can make the three expressions squares     Can you find three distinct positive integers that satisfy the system of equations?                     … Continue reading

Posted in Number Puzzles | Tagged | Leave a comment