Monthly Archives: June 2016

Each of a-b, a+n, b+n, a+b+n is a square

    To find two numbers,     and   ,   whose difference,     is a square and such that each,     and   ,   is a square and their sum,   ,   is a … Continue reading

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Puzzle – geometrical progression

      To find 3 numbers,   ,   in geometrical progression such that each increased by a given number     is a square.                             … Continue reading

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Integer n such sum of its aliquot divisors is a square

  Some numbers never come up as aliquot sums,   such as 5. They are called nonaliquot numbers or untouchable numbers https://en.wikipedia.org/wiki/Untouchable_number       Let’s find the lowest     such that   ,       :    … Continue reading

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Sums of consecutive integers that result in a 4-th power

  1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 … Continue reading

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Puzzle #2 – Sum of aliquot divisors of a and b

    Find pair of integers (a, b) with the property that the sum of the aliquot divisors of a exceeds the sum of the aliquot divisors of by a square number.   Here are two examples:   99   … Continue reading

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Puzzle – Sum of aliquot divisors of n

    The aliquot divisors of   n   are the divisors of   n   less than   n     Find two squares such that if each is increased by the sum of its aliquot divisors the resulting … Continue reading

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n-digit numbers whose 4th powers end with the same n digits

                                                                                                                                      …………………………………………………………… ……………………………………………………………                                                                                                           ……………………………………………………….. ………………………………………………………..               … Continue reading

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n-digit numbers whose cubes end with the same n digits

    1-digit :   =   64   =   216   =   729   2-digit :   =   13824   =   15625   =   117649   =   438976   3-digit :   = … Continue reading

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When sum of reciprocal squares = 1/2

    9 terms:   10 terms:   11 terms:     Can you express   1/2   with less than 9 terms?                                  

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8-digit numbers whose squares end with the same 8 digits

        =   7588043387109376   =   166168212890625   A general form:    numbers whose squares end with the same… … 2 digits: … 3 digits: … 4 digits: … 5 digits: … 6 digits: … 7 … Continue reading

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