Pythagorean triangles diff. between sides,perim,diam. of inscribed circles are squares,diff. between Areas a cube

 
 
Find two Pythagorean triangles   (a_1, \; b_1, \; c_1)   and   (a_2, \; b_2, \; c_2)   such that

a_1 \; - \; a_2
b_1 \; - \; b_2
c_1 \; - \; c_2
p_1 \; - \; p_2
d_1 \; - \; d_2

are all squares

And, the difference of areas a cube

p_1, \; p_2   represent the respective perimeters
d_1, \; d_2   the respective diameters of inscribed circles

 
 
Unfortunately, Paul’s solution do not satisfy all conditions mentioned.   The difference of perimeters should also be a square.

(2880, 14256, 14544),   (2304, 9072, 9360)
31680 – 20736   =   10944   =   2^6 \times 3^2 \times 19

(2754, 23328, 23490),   (2430, 18144, 18306)
49572 – 38880   =   10692   =   2^2 \times 3^5 \times 11

(2720, 28836, 28964),   (2464, 23652, 23780)
60520 – 49896   =   10624   =   2^7 \times 83

(2664, 49248, 49320),   (2520, 44064, 44136)
101232 – 90720   =   10512   =   2^4 \times 3^2 \times 73

(2624, 107568, 107600),   (2560, 102384, 102416)
217792 – 207360   =   10432   =   2^6 \times 163

(2610, 189216, 189234),   (2574, 184032, 184050)
381060 – 370656   =   10404   =   102^2

(2600, 422496, 422504),   (2584, 417312, 417320)
847600 – 837216   =   10384   =   2^4 \times 11 \times 59

(2594, 1682208, 1682210),   (2590, 1677024, 1677026)
3367012 – 3356640   =   10372   =   2^2 \times 2593

 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 

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About benvitalis

math grad - Interest: Number theory
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5 Responses to Pythagorean triangles diff. between sides,perim,diam. of inscribed circles are squares,diff. between Areas a cube

  1. Paul says:

    Here are a few that match all the criteria. Format is :-
    {{a1, b1, c1},{a2, b2, c2}}, diff inscribed circle diameters.

    {{2304,9072,9360},{2880,14256,14544}} , 24^2
    {{2430,18144,18306},{2754,23328,23490}} , 18^2
    {{2464,23652,23780},{2720,28836,28964}} , 16^2
    {{2520,44064,44136},{2664,49248,49320}} , 12^2
    {{2560,102384,102416},{2624,107568,107600}} , 8^2
    {{2574,184032,184050},{2610,189216,189234}} , 6^2
    {{2584,417312,417320},{2600,422496,422504}} , 4^2
    {{2590,1677024,1677026},{2594,1682208,1682210}} , 2^2

    {{2304,9072,9360},{2880,14256,14544}}
    2880 – 2304 = 24^2
    14256 – 9072 = 72^2
    14544 – 9360 = 72^2
    (2880 * 14256)/2 – (2304 * 9072)/2 = 216^3
    2592 – 2016 = 24^2

    {{2430,18144,18306},{2754,23328,23490}}
    2754 – 2430 = 18^2
    23328 – 18144 = 72^2
    23490 – 18306 = 72^2
    (2754 * 23328)/2 – (2430 * 18144)/2 = 216^3
    2592 – 2268 = 18^2

    {{2464,23652,23780},{2720,28836,28964}}
    2720 – 2464 = 16^2
    28836 – 23652 = 72^2
    28964 – 23780 = 72^2
    (2720 * 28836)/2 – (2464 * 23652)/2 = 216^3
    2592 – 2336 = 16^2

    {{2520,44064,44136},{2664,49248,49320}}
    2664 – 2520 = 12^2
    49248 – 44064 = 72^2
    49320 – 44136 = 72^2
    (2664 * 49248)/2 – (2520 * 44064)/2 = 216^3
    2592 – 2448 = 12^2

    {{2560,102384,102416},{2624,107568,107600}}
    2624 – 2560 = 8^2
    107568 – 102384 = 72^2
    107600 – 102416 = 72^2
    (2624 * 107568)/2 – (2560 * 102384)/2 = 216^3
    2592 – 2528 = 8^2

    {{2574,184032,184050},{2610,189216,189234}}
    2610 – 2574 = 6^2
    189216 – 184032 = 72^2
    189234 – 184050 = 72^2
    (2610 * 189216)/2 – (2574 * 184032)/2 = 216^3
    2592 – 2556 = 6^2

    {{2584,417312,417320},{2600,422496,422504}}
    2600 – 2584 = 4^2
    422496 – 417312 = 72^2
    422504 – 417320 = 72^2
    (2600 * 422496)/2 – (2584 * 417312)/2 = 216^3
    2592 – 2576 = 4^2

    {{2590,1677024,1677026},{2594,1682208,1682210}}
    2594 – 2590 = 2^2
    1682208 – 1677024 = 72^2
    1682210 – 1677026 = 72^2
    (2594 * 1682208)/2 – (2590 * 1677024)/2 = 216^3
    2592 – 2588 = 2^2

    Paul

  2. Paul says:

    Hey, don’t rush for me, get yourself better.

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