Num3er 64 – DigitSum(64) & DigitProduct(64)

 
 
True or False:

64   is the only square the sum of whose digits less unity and product plus unity are squares

64 \; = \; 8^2
(6+4) \; - \; 1 \; = \; 3^2
(6\times 4) \; + \; 1 \; = \; 5^2

and,   3 + 5 = 8

 

 
 

 
 

 
 

 
 
 
 
 

 
 

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About benvitalis

math grad - Interest: Number theory
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3 Responses to Num3er 64 – DigitSum(64) & DigitProduct(64)

  1. pipo says:

    False: 433 is the next number, which square follows the rules.
    433^2 = 187489
    (1+8+7+4+8+9) -1 = 36 = 6^2
    (1*8*7*4*8*9)+1 = 16129 = 127^2
    Other numbers below 100000 are: 728, 1727, 2483, 6769, 7435, 7721, 9269, 9442, 10691,10882, 13463,13535, 14966,15227, 20278, 21131, 24139, 24479, 25381, 29611, 30689, 34127, 35531, 38161, 40211, 40807, 42911, 47161, 50129, 52111, 57815, 59489, 74312, 90073, 97579
    pipo

    • Paul says:

      I found a lot like that Pipo, but in your first example 6 + 127 is not equal to 433, ( the last statement). I went to 1,000,000 and found none. This is the last one I found

      {{3,6}, {1,6,1,2,9}, 6, 127, 133, 901189}

      the 133 has to equal 901189 and is getting further behind at each part solution, I suspect there are no other solutions that meet all criteria.

      Paul.

  2. pipo says:

    I overlooked the last criterium, so indeed you are right.
    pipo

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