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

 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 

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Pythagorean triangle-perimeter a square and diameter of the inscribed circle a cube

 

 
Find a Pythagorean triangle with perimeter   p   a square and diameter of the inscribed circle a cube

Let the sides be

(m^2 - n^2) \,x,     2 \, m \, n \, x,     (m^2 + n^2) \,x

p \; = \; 2 \, m \, x \, (m + n) \; = \; r^2

Then     r \; = \;  n \, x \, (m - n)

the diameter   d   is to be a cube,   say   r^3 \,/ \,s^3

d \; = \; 2 \, n \, x \, (m - n) \; = \; r^3 \,/ \,s^3

From the two values of   x

x \; = \; r^2 \,/ \,(2 \, m \, (m+n))
x \; = \; r^3 \,/ \,(2 \, n \, s^3 \, (m-n))

we get   r   in terms of   m, n, s
r^2 \,/ \,(2 \, m \, (m+n)) \; = \; r^3 \,/ \,(2 \, n \, s^3 \, (m-n))

r \; = \; (m \, s^3 \, (n-m)) \,/ \,(n \, (m+n))

 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 

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2 Pythagorean triangles – sum/difference of perimeters,difference of areas are squares

 
 

Find two Pythagorean triangles the sum or difference of whose perimeters is a square,
the difference of areas a square

 

 
 

 
 

 
 
 
 
 

 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 

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Pythagorean triangles – each side is a sum of two squares

 
 

Find Pythagorean triangles each of whose sides is a sum of two squares.

For example,     (9, \; 40, \; 41)

9 \; = \; 3^2
40 \; = \; 2^2 \; + \; 6^2
41 \; = \; 4^2 \; + \; 5^2

 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 

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(A_1)^n + (A_2)^n +…+ (A_5)^n = (B_1)^n +…+ (B_5)^n, n = 1,3,5,7

 
 

(A_1)^n + (A_2)^n + (A_3)^n + (A_4)^n + (A_5)^n = (B_1)^n + (B_2)^n + (B_3)^n + (B_4)^n + (B_5)^n

where   n = 1, 3, 5, 7
 

5 + 55 + 65 + 125 + 127 = 13 + 37 + 77 + 119 + 131
5^3 + 55^3 + 65^3 + 125^3 + 127^3 = 13^3 + 37^3 + 77^3 + 119^3 + 131^3
5^5 + 55^5 + 65^5 + 125^5 + 127^5 = 13^5 + 37^5 + 77^5 + 119^5 + 131^5
5^7 + 55^7 + 65^7 + 125^7 + 127^7 = 13^7 + 37^7 + 77^7 + 119^7 + 131^7

7 + 263 + 361 + 639 + 649 = 33 + 207 + 399 + 613 + 667
7^3 + 263^3 + 361^3 + 639^3 + 649^3 = 33^3 + 207^3 + 399^3 + 613^3 + 667^3
7^5 + 263^5 + 361^5 + 639^5 + 649^5 = 33^5 + 207^5 + 399^5 + 613^5 + 667^5
7^7 + 263^7 + 361^7 + 639^7 + 649^7 = 33^7 + 207^7 + 399^7 + 613^7 + 667^7

13 + 59 + 67 + 131 + 141 = 27 + 33 + 81 + 127 + 143
13^3 + 59^3 + 67^3 + 131^3 + 141^3 = 27^3 + 33^3 + 81^3 + 127^3 + 143^3
13^5 + 59^5 + 67^5 + 131^5 + 141^5 = 27^5 + 33^5 + 81^5 + 127^5 + 143^5
13^7 + 59^7 + 67^7 + 131^7 + 141^7 = 27^7 + 33^7 + 81^7 + 127^7 + 143^7

13 + 307 + 407 + 737 + 743 = 47 + 233 + 457 + 703 + 767
13^3 + 307^3 + 407^3 + 737^3 + 743^3 = 47^3 + 233^3 + 457^3 + 703^3 + 767^3
13^5 + 307^5 + 407^5 + 737^5 + 743^5 = 47^5 + 233^5 + 457^5 + 703^5 + 767^5
13^7 + 307^7 + 407^7 + 737^7 + 743^7 = 47^7 + 233^7 + 457^7 + 703^7 + 767^7

17 + 63 + 87 + 133 + 137 = 33 + 37 + 107 + 117 + 143
17^3 + 63^3 + 87^3 + 133^3 + 137^3 = 33^3 + 37^3 + 107^3 + 117^3 + 143^3
17^5 + 63^5 + 87^5 + 133^5 + 137^5 = 33^5 + 37^5 + 107^5 + 117^5 + 143^5
17^7 + 63^7 + 87^7 + 133^7 + 137^7 = 33^7 + 37^7 + 107^7 + 117^7 + 143^7

17 + 71 + 103 + 157 + 163 = 31 + 47 + 121 + 143 + 169
17^3 + 71^3 + 103^3 + 157^3 + 163^3 = 31^3 + 47^3 + 121^3 + 143^3 + 169^3
17^5 + 71^5 + 103^5 + 157^5 + 163^5 = 31^5 + 47^5 + 121^5 + 143^5 + 169^5
17^7 + 71^7 + 103^7 + 157^7 + 163^7 = 31^7 + 47^7 + 121^7 + 143^7 + 169^7

19 + 61 + 77 + 119 + 121 = 29 + 41 + 97 + 103 + 127
19^3 + 61^3 + 77^3 + 119^3 + 121^3 = 29^3 + 41^3 + 97^3 + 103^3 + 127^3
19^5 + 61^5 + 77^5 + 119^5 + 121^5 = 29^5 + 41^5 + 97^5 + 103^5 + 127^5
19^7 + 61^7 + 77^7 + 119^7 + 121^7 = 29^7 + 41^7 + 97^7 + 103^7 + 127^7

19 + 101 + 157 + 239 + 251 = 31 + 79 + 173 + 227 + 257
19^3 + 101^3 + 157^3 + 239^3 + 251^3 = 31^3 + 79^3 + 173^3 + 227^3 + 257^3
19^5 + 101^5 + 157^5 + 239^5 + 251^5 = 31^5 + 79^5 + 173^5 + 227^5 + 257^5
19^7 + 101^7 + 157^7 + 239^7 + 251^7 = 31^7 + 79^7 + 173^7 + 227^7 + 257^7

19 + 121 + 131 + 221 + 269 = 41 + 71 + 169 + 209 + 271
19^3 + 121^3 + 131^3 + 221^3 + 269^3 = 41^3 + 71^3 + 169^3 + 209^3 + 271^3
19^5 + 121^5 + 131^5 + 221^5 + 269^5 = 41^5 + 71^5 + 169^5 + 209^5 + 271^5
19^7 + 121^7 + 131^7 + 221^7 + 269^7 = 41^7 + 71^7 + 169^7 + 209^7 + 271^7

27 + 123 + 147 + 245 + 295 = 55 + 71 + 179 + 235 + 297
27^3 + 123^3 + 147^3 + 245^3 + 295^3 = 55^3 + 71^3 + 179^3 + 235^3 + 297^3
27^5 + 123^5 + 147^5 + 245^5 + 295^5 = 55^5 + 71^5 + 179^5 + 235^5 + 297^5
27^7 + 123^7 + 147^7 + 245^7 + 295^7 = 55^7 + 71^7 + 179^7 + 235^7 + 297^7

29 + 251 + 253 + 431 + 529 = 67 + 149 + 341 + 403 + 533
29^3 + 251^3 + 253^3 + 431^3 + 529^3 = 67^3 + 149^3 + 341^3 + 403^3 + 533^3
29^5 + 251^5 + 253^5 + 431^5 + 529^5 = 67^5 + 149^5 + 341^5 + 403^5 + 533^5
29^7 + 251^7 + 253^7 + 431^7 + 529^7 = 67^7 + 149^7 + 341^7 + 403^7 + 533^7

33 + 97 + 103 + 167 + 183 = 47 + 63 + 133 + 153 + 187
33^3 + 97^3 + 103^3 + 167^3 + 183^3 = 47^3 + 63^3 + 133^3 + 153^3 + 187^3
33^5 + 97^5 + 103^5 + 167^5 + 183^5 = 47^5 + 63^5 + 133^5 + 153^5 + 187^5
33^7 + 97^7 + 103^7 + 167^7 + 183^7 = 47^7 + 63^7 + 133^7 + 153^7 + 187^7

35 + 121 + 161 + 247 + 253 = 65 + 71 + 205 + 211 + 265
35^3 + 121^3 + 161^3 + 247^3 + 253^3 = 65^3 + 71^3 + 205^3 + 211^3 + 265^3
35^5 + 121^5 + 161^5 + 247^5 + 253^5 = 65^5 + 71^5 + 205^5 + 211^5 + 265^5
35^7 + 121^7 + 161^7 + 247^7 + 253^7 = 65^7 + 71^7 + 205^7 + 211^7 + 265^7

35 + 161 + 241 + 367 + 383 = 61 + 115 + 275 + 341 + 395
35^3 + 161^3 + 241^3 + 367^3 + 383^3 = 61^3 + 115^3 + 275^3 + 341^3 + 395^3
35^5 + 161^5 + 241^5 + 367^5 + 383^5 = 61^5 + 115^5 + 275^5 + 341^5 + 395^5
35^7 + 161^7 + 241^7 + 367^7 + 383^7 = 61^7 + 115^7 + 275^7 + 341^7 + 395^7

37 + 229 + 233 + 481 + 507 = 81 + 129 + 297 + 463 + 517
37^3 + 229^3 + 233^3 + 481^3 + 507^3 = 81^3 + 129^3 + 297^3 + 463^3 + 517^3
37^5 + 229^5 + 233^5 + 481^5 + 507^5 = 81^5 + 129^5 + 297^5 + 463^5 + 517^5
37^7 + 229^7 + 233^7 + 481^7 + 507^7 = 81^7 + 129^7 + 297^7 + 463^7 + 517^7

49 + 147 + 171 + 269 + 271 = 59 + 117 + 201 + 247 + 283
49^3 + 147^3 + 171^3 + 269^3 + 271^3 = 59^3 + 117^3 + 201^3 + 247^3 + 283^3
49^5 + 147^5 + 171^5 + 269^5 + 271^5 = 59^5 + 117^5 + 201^5 + 247^5 + 283^5
49^7 + 147^7 + 171^7 + 269^7 + 271^7 = 59^7 + 117^7 + 201^7 + 247^7 + 283^7

49 + 151 + 209 + 343 + 407 = 85 + 95 + 235 + 335 + 409
49^3 + 151^3 + 209^3 + 343^3 + 407^3 = 85^3 + 95^3 + 235^3 + 335^3 + 409^3
49^5 + 151^5 + 209^5 + 343^5 + 407^5 = 85^5 + 95^5 + 235^5 + 335^5 + 409^5
49^7 + 151^7 + 209^7 + 343^7 + 407^7 = 85^7 + 95^7 + 235^7 + 335^7 + 409^7

49 + 331 + 539 + 821 + 869 = 71 + 289 + 569 + 799 + 881
49^3 + 331^3 + 539^3 + 821^3 + 869^3 = 71^3 + 289^3 + 569^3 + 799^3 + 881^3
49^5 + 331^5 + 539^5 + 821^5 + 869^5 = 71^5 + 289^5 + 569^5 + 799^5 + 881^5
49^7 + 331^7 + 539^7 + 821^7 + 869^7 = 71^7 + 289^7 + 569^7 + 799^7 + 881^7

65 + 399 + 639 + 973 + 1027 = 99 + 335 + 685 + 939 + 1045
65^3 + 399^3 + 639^3 + 973^3 + 1027^3 = 99^3 + 335^3 + 685^3 + 939^3 + 1045^3
65^5 + 399^5 + 639^5 + 973^5 + 1027^5 = 99^5 + 335^5 + 685^5 + 939^5 + 1045^5
65^7 + 399^7 + 639^7 + 973^7 + 1027^7 = 99^7 + 335^7 + 685^7 + 939^7 + 1045^7

69 + 151 + 191 + 299 + 309 = 99 + 101 + 221 + 279 + 319
69^3+151^3+191^3+299^3+309^3 = 99^3+101^3+221^3+279^3+319^3
69^5+151^5+191^5+299^5+309^5 = 99^5+101^5+221^5+279^5+319^5
69^7+151^7+191^7+299^7+309^7 = 99^7+101^7+221^7+279^7+319^7

69 + 261 + 333 + 551 + 659 = 137 + 149 + 391 + 533 + 663
69^3 + 261^3 + 333^3 + 551^3 + 659^3 = 137^3 + 149^3 + 391^3 + 533^3 + 663^3
69^5 + 261^5 + 333^5 + 551^5 + 659^5 = 137^5 + 149^5 + 391^5 + 533^5 + 663^5
69^7 + 261^7 + 333^7 + 551^7 + 659^7 = 137^7 + 149^7 + 391^7 + 533^7 + 663^7

83 + 257 + 313 + 487 + 493 = 113 + 187 + 383 + 433 + 517
83^3 + 257^3 + 313^3 + 487^3 + 493^3 = 113^3 + 187^3 + 383^3 + 433^3 + 517^3
83^5 + 257^5 + 313^5 + 487^5 + 493^5 = 113^5 + 187^5 + 383^5 + 433^5 + 517^5
83^7 + 257^7 + 313^7 + 487^7 + 493^7 = 113^7 + 187^7 + 383^7 + 433^7 + 517^7

91 + 269 + 301 + 479 + 481 = 101 + 229 + 341 + 451 + 499
91^3 + 269^3 + 301^3 + 479^3 + 481^3 = 101^3 + 229^3 + 341^3 + 451^3 + 499^3
91^5 + 269^5 + 301^5 + 479^5 + 481^5 = 101^5 + 229^5 + 341^5 + 451^5 + 499^5
91^7 + 269^7 + 301^7 + 479^7 + 481^7 = 101^7 + 229^7 + 341^7 + 451^7 + 499^7

91 + 301 + 389 + 599 + 611 = 151 + 191 + 499 + 509 + 641
91^3 + 301^3 + 389^3 + 599^3 + 611^3 = 151^3 + 191^3 + 499^3 + 509^3 + 641^3
91^5 + 301^5 + 389^5 + 599^5 + 611^5 = 151^5 + 191^5 + 499^5 + 509^5 + 641^5
91^7 + 301^7 + 389^7 + 599^7 + 611^7 = 151^7 + 191^7 + 499^7 + 509^7 + 641^7

91 + 453 + 693 + 1055 + 1105 = 153 + 341 + 775 + 993 + 1135
91^3 + 453^3 + 693^3 + 1055^3 + 1105^3 = 153^3 + 341^3 + 775^3 + 993^3 + 1135^3
91^5 + 453^5 + 693^5 + 1055^5 + 1105^5 = 153^5 + 341^5 + 775^5 + 993^5 + 1135^5
91^7 + 453^7 + 693^7 + 1055^7 + 1105^7 = 153^7 + 341^7 + 775^7 + 993^7 + 1135^7

95 + 295 + 395 + 715 + 793 = 163 + 187 + 443 + 701 + 799
95^3 + 295^3 + 395^3 + 715^3 + 793^3 = 163^3 + 187^3 + 443^3 + 701^3 + 799^3
95^5 + 295^5 + 395^5 + 715^5 + 793^5 = 163^5 + 187^5 + 443^5 + 701^5 + 799^5
95^7 + 295^7 + 395^7 + 715^7 + 793^7 = 163^7 + 187^7 + 443^7 + 701^7 + 799^7

 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 

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2 Pythagorean triangles (a,b,c),(d,e,f); a-b=e-f and b-c=d-e

 
 

Find two Pythagorean triangles   (a, b, c)   and   (d, e, f)   for which

a – b = e – f,   and
b – c = d – e
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 

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Pythagorean triangle {a+b+c, a^2+b+c, a+(b+c)^2} are to made squares

 

 
Find a Pythagorean triangle   (a, \; b, \; c)   such that

a \; + \; b \; + \; c
a^2 \; + \; b \; + \; c
b^2 \; + \; c \; + \; a
c^2 \; + \; a \; + \; b
a \; + \; (b+c)^2
b \; + \; (c+a)^2
c \; + \; (a+b)^2

are all squares
 
 
 
 
 
 
 
 
 
 

 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 

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Pythagorean triangle (a, b, c): {a^2 – a, b^2 – b, c^2 – c} to be made squares

 
 

Find a Pythagorean triangle   (a, \; b, \; c)   such that the square of any side exceeds that side by a square

a^2 \; - \; a \; = \; m^2
b^2 \; - \; b \; = \; n^2
c^2 \; - \; c \; = \; p^2

 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 

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Pythagorean triangle-perimeter + square of any side is a square

 
 

Find a Pythagorean triangle the sum whose perimeter and square of any side is a square

 
 

Let the sides be   a \,x, \; b \,x, \; c \,x,   where   a^2 + b^2 = c^2

Then

x^2 \; + \; p \,x
x^2 \; + \; q \,x
x^2 \; + \; r \,x

are made squares

where    p = s/a^2,    q = s/b^2,    r = s/c^2,
                 s = a + b + c

 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 

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Pythagorean triangles {a^2 + b^2 ± a*b/2, a^2 + b^2 ± a*b} to be made squares

 
 
Solve

(1)

(a^2 + b^2) \; - \; a \, b \; = \; n^2
(a^2 + b^2) \; + \; a \, b \; = \; m^2

(2)

(a^2 + b^2) \; - \; 1/2 \, \cdot \, a \, b \; = \; p^2
(a^2 + b^2) \; + \; 1/2 \, \cdot \, a \, b \; = \; q^2

 

where   (a, \; b)   are the two odd legs in a Pythagorean triangle.

 

 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 

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