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Triangular & Second pentagonal numbers

  Suppose the positive integers     satisfy   ………. (1) Or     is a Triangular number  [     ] numbers of the form     are Second pentagonal numbers.   Equation   (1)   gives us the … Continue reading

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Alphametic puzzles — Part 2

  (1)   are positive integers in arithmetic sequence such that ,     and     Find     (2) Substitute each of the letters with a different decimal digit from 0 to 9, to satisfy the following system of simultaneous … Continue reading

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Positive integers (a,b,c) in AP; a+b, b+c, c+a are squares

    are in arithmetic progression   Here are some solutions:                              

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Pythagorean triples (x^2 – 1, y^2 – 1, z^2 – 1)

    To find a Pythagorean triangle with sides, where     are integers.   Here are two examples,                                

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Concatenation: 2016 || x || 2016 = y^3

  Previous blog:   Concatenation: 2016 || x || 2016 = y^2   Paul found: ………………..     Pipo found: A few more, 2016 appear every 2500 starting at 856                     … Continue reading

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a^n + b^n + c^n = x^n + y^n + z^n, n = 2, 4

  Find six distinct positive integers     such that     for example,     …..     Paul found:   …     …     …     …     …     …     … Continue reading

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Open question: Positive integers as a sum of squares and cubes?

    Every positive integer is a sum of four squares (Lagrange’s Theorem) It is known that every positive integer is a sum of no more than 9 positive cubes, and that every “sufficiently large” integer is a sum of … Continue reading

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123456789+987654321

  777777   has   48 divisors: 1   3   7   11   13   21   33   37   39   49   77   91   111   143   147   231   259   … Continue reading

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Integers that can be written in base n^2 + 1with the same digits but in opposite order

  Show that integers of the form     and   can be written in base     with the same digits but in opposite order                             … Continue reading

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N^5 = A^2 + B^2 for N ≤ 1000

      and     are non-zero positive integers   multiplying both sides by   When is the sum of two squares a cube? https://benvitalenum3ers.wordpress.com/2015/05/06/when-is-the-sum-of-two-squares-a-cube/ multiplying both sides by     Solutions when   a = 1     … Continue reading

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