Pos+ Integers (x,y); x + y = a^2 and x*y = b^2

 
 
To find   (x, \; y)   positive integers such that

x \; < \; y ,     and

x \; + \; y \; = \; a^2
x \, y \; = \; b^2

 

2 \; + \; 2 \; = \; 4 \; = \; 2^2
2 \; \times \; 2 \; = \; 4

5 \; + \; 20 \; = \; 25 \; = \; 5^2
5 \; \times \; 20 \; = \; 100 \; = \; 10^2

10 \; + \; 90 \; = \; 100 \; = \; 10^2
10 \; \times \; 90 \; = \; 900 \; = \; 30^2

17 \; + \; 272 \; = \; 289 \; = \; 17^2
17 \; \times \; 272 \; = \; 4624 \; = \; 68^2

26 \; + \; 650 \; = \; 676 \; = \; 26^2
26 \; \times \; 650 \; = \; 16900 \; = 130^2

 
Note that

The sequence for the values of   x :    2,   5,   10,   17,   26

and

2
5   –   2   =   3
10   –   5   =   5
17   –   10   =   7
26   –   17   =   9
…………
…………??

 
Does the sequence continue?

Yes, it does.

 
                                 —————————————————-       
 
 

Paul found:

37 \; + \; 1332 \; = \; 1369 \; = \; 37^2
37 \; \times \; 1332 \; = \; 49284 \; = \; 222^2

50 \; + \; 2450 \; = \; 2500 \; = \; 50^2
50 \; \times \; 2450 \; = \; 122500 \; = \; 350^2

65 \; + \; 4160 \; = \; 4225 \; = \; 65^2
65 \; \times \; 4160 \; = \; 270400 \; = \; 520^2

82 \; + \; 6642 \; = \; 6724 \; = \; 82^2
82 \; \times \; 6642 \; = \; 544644 \; = \; 738^2

101 \; + \; 10100 \; = \; 10201 \; = \; 101^2
101 \; \times \; 10100 \; = \; 1020100 \; = \; 1010^2

122 \; + \; 14762 \; = \; 14884 \; = \; 122^2
122 \; \times \; 14762 \; = \; 1800964 \; = \; 1342^2

145 \; + \; 20880 \; = \; 21025 \; = \; 145^2
145 \; \times \; 20880 \; = \; 3027600 \; = \; 1740^2

170 \; + \; 28730 \; = \; 28900 \; = \; 170^2
170 \; \times \; 28730 \; = \; 4884100 \; = \; 2210^2

197 \; + \; 38612 \; = \; 38809 \; = \; 197^2
197 \; \times \; 38612 \; = \; 7606564 \; = \; 2758^2

226 \; + \; 50850 \; = \; 51076 \; = \; 226^2
226 \; \times \; 50850 \; = \; 11492100 \; = \; 3390^2

37 \; - \; 26 \; = \; 11
50 \; - \; 37 \; = \; 13
65 \; - \; 50 \; = \; 15
82 \; - \; 65 \; = \; 17
101 \; - \; 82 \; = \; 19
122 \; - \; 101 \; = \; 21
145 \; - \; 122 \; = \; 23
170 \; - \; 145 \; = \; 25
197 \; - \; 170 \; = \; 27
226 \; - \; 197 \; = \; 29

The generating functions for x and y are:
x \; = \; n^2 \; + \; 1
y \; = \; n^2 \,(n^2+1)

x \; + \; y \; = \; (1 + n^2)^2
x \, y \; = \; (n + n^3)^2

 
 
 
 
 
 
 
 
 
 
 
 
 
 
 

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About benvitalis

math grad - Interest: Number theory
This entry was posted in Number Puzzles and tagged . Bookmark the permalink.

One Response to Pos+ Integers (x,y); x + y = a^2 and x*y = b^2

  1. paul says:

    Yes it does. The generating functions for x and y are:-
    n^2+1 and n^2(n^2+1)
    These then form the following generating functions for the squares of
    x + y = (1 + n^2)^2 and
    x y = (n + n^3)^2

    Here are the next 10 in the sequence format {x, y, a^2, b^2}

    37, 1332, 1369, 49284
    50, 2450, 2500, 122500
    65, 4160, 4225, 270400
    82, 6642, 6724, 544644
    101, 10100, 10201, 1020100
    122, 14762, 14884, 1800964
    145, 20880, 21025, 3027600
    170, 28730, 28900, 4884100
    197, 38612, 38809, 7606564
    226, 50850, 51076, 11492100

    Paul.

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