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

Answer 1

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.

Answer 2

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)