## 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   =   $2^2 \times 2593$

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

• benvitalis says:

Sorry Paul, I’ve been sick confined to bed. I’m getting better. I’ll be back to respond

• benvitalis says:

Not all the criteria.
The difference between perimeters should also be a square

2. Paul says:

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

• benvitalis says:

Tks. I feel better. I”ll be back on this site Tomorrow