Number Theory.,Math

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How many ordered pairs of positive integers (a,b) are there such that (4a-1)/b and (4b-1)/a are both integers?

  • Number Theory.,Math -

    clearly (1,1) will work

    I then made up and ran a silly little computer program where I let a go from 1 to 1000 and b = 1 to 1000 and tested each case
    (so my program tested 1 million cases)
    it found the following ordered pairs

    1 3 3 1 5 19 7 9 9 7 19 5

    testing (19,5)
    if a = 19 and b=5
    (4a-1)/b = 75/5 = 15 which is an integer
    (4b-1)/a = 19/19 = 1 , which is an integer

    notice they come in symmetric pairs
    so there 7 of these.

  • Number Theory.,Math -

    I just changed my a values from 1 to 10,000
    and my b value from 1 to 10,000
    and it found no others.

    should have put them in ordered pairs:

    (1,1) (1,3) (3,1) (5,19) , (7,9), (9,7), (19,5)

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