Is there a pair of integers (a,b) such that a,x1,y1,b is part of an arithmetic sequences and a,x2,y2,b is part of a geometric sequence with x1,x2,y1,y2 all integers?

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Jim Hunt · Accepted Answer

It is given that x1,x2,y1,y2 are all integers. a,x1,y1,b is part of an arithmetic sequences. Thus, x1−a=y1−x1 the common difference is same in arithmetic sequences. x1−a=y1−x1 2x1=y1+a a=2x1−y1 So, a is integer as x1 and y1 are integers. (1) y1−x1=b−y1 the common difference is same in arithmetic sequences. y1−x1=b−y1 2y1=x1+b b=2y1−x1 So, b is integer as x1 and y1 are integers. (2) It is given that a,x2,y2,b is part of a geometric sequence. Hence, x2a=y2x2 the common ratio is same in geometric sequence. x2x2=ay2 a=x22y2 x2=ay2 Similarly, y2=bx2 Also, x2a=by2 the common ratio is same in geometric sequence. ab=x2y2 ab is an integer as x2 and y2 are integerand the product of two integer is an integer. Thus, from equations it is proved that there is a pair of integers (a,b).

Is there a pair of integers (a, b) such that

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