Find two distinct positive integers and such that

Each of and is an integer.

Here are the first few solutions:

For all members of the sequence, is a square

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Paul answered correctly to the question:

Is there an infinite numbers of pairs ?

The square of each member of the sequence is a Centered 12-gonal numbers,

also known as, Star numbers.

that is,

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I do believe there is another pair sequence between those given.

There is a recurrence of {10, -1} from one to the next and so there will be an infinite number of pairs.

(x, y) = (1, 11)

(1^2 – 12)/11 = -1 ….. (11^2 – 12)/1 = 109

(x, y) = (11, 109)

(11^2 – 12)/109 = 1 ….. (109^2 – 12)/11 = 1079

(x, y) = (109, 1079)

(109^2 – 12)/1079 = 11 ….. (1079^2 – 12)/109 = 10681

(x, y) = (1079, 10681)

(1079^2 – 12)/10681 = 109 ….. (10681^2 – 12)/1079 = 105731

(x, y) = (10681, 105731)

(10681^2 – 12)/105731 = 1079 ….. (105731^2 – 12)/10681 = 1046629

(x, y) = (105731, 1046629)

(105731^2 – 12)/1046629 = 10681 ….. (1046629^2 – 12)/105731 = 10360559

(x, y) = (1046629, 10360559)

(1046629^2 – 12)/10360559 = 105731 ….. (10360559^2 – 12)/1046629 = 102558961

(x, y) = (10360559, 102558961)

(10360559^2 – 12)/102558961 = 1046629 ….. (102558961^2 – 12)/10360559 = 1015229051

(x, y) = (102558961, 1015229051)

(102558961^2 – 12)/1015229051 = 10360559 ….. (1015229051^2 – 12)/102558961 = 10049731549

(x, y) = (1015229051, 10049731549)

(1015229051^2 – 12)/10049731549 = 102558961 ….. (10049731549^2 – 12)/1015229051 = 99482086439

Paul.

You are right. I missed a sequence of numbers

I added some stuff