Part 2 – PT diff. between sides,perim,diam. of inscribed circles are squares,diff. between Areas a cube

 
 
Part 1 – 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

 
 
Take, for example, the triples of the forms:

(a_1, \; b_1, \; c_1) \; = \; (10 \, x^2, \; 24 \, x^2, \; 26 \, x^2)
(a_2, \; b_2, \; c_2) \; = \; (6 \, x^2, \; 8 \, x^2, \; 10 \, x^2)

a_1 - a_2 = 10 \, x^2 - 6 \, x^2 \; = \; 4 \, x^2 \; = \; (2 \, x)^2
b_1 - b_2 = 24 \, x^2 - 8 \, x^2 \; = \; 16 \, x^2 \; = \; (4 \, x)^2
c_1 - c_2 = 26 \, x^2 - 10 \, x^2 \; = \; 16 \, x^2 \; = \; (4 \, x)^2

Perimeters:
P_1 = a_1 + b_1 + c_1 = 10 \, x^2 + 24 \, x^2 + 26 \, x^2 \; = \; 60 \, x^2
P_2 = a_2 + b_2 + c_2 = 6 \, x^2 + 8 \, x^2 + 10 \, x^2 \; = \; 24 \, x^2

Difference of perimeters:
P_1 - P_2 = 60 \, x^2 - 24 \, x^2 \; = \; 36 \, x^2 \; = \; (6 \, x)^2

Diameter of inscribed circle:
d_1 = a_1 + b_1 - c_1 = 10 \, x^2 + 24 \, x^2 - 26 \, x^2 \; = \; 8 \, x^2
d_2 = a_2 + b_2 - c_2 = 6 \, x^2 + 8 \, x^2 - 10 \, x^2 \; = \; 4 \, x^2 \; = \; (2 \, x)^2

Difference of diameters:
d_1 - d_2 = 8 \, x^2 - 4 \, x^2 \; = \; 4 \, x^2 \; = \; (2 \, x)^2

Areas:
A_1 = 1/2 \; \times \; (10 \, x^2) \,(24 \, x^2) \; = \; 120 \, x^4
A_2 = 1/2 \; \times \; (6 \, x^2) \,(8 \, x^2) \; = \; 24 \, x^4

A_1 - A_2 = 120 \, x^4 - 24 \, x^4 \; = \; 96 \, x^4

The difference   96 \, x^4   of the areas is a cube if   x \; = \; p^3/12

96 \, (p^3 \,/ \,12)^4 \; = \; p^{12} \,/ \,216

 

 
 
   x   …..   Difference of areas
                          A_1 - A_2
——————————————————————
     18   …..   10077696 = 6^9
   144   …..   41278242816 = 3456^3
   486   …..   5355700839936 = 17496^3
  1152   …..   169075682574336 = 55296^3
  2250   …..   2460375000000000 = 135000^3
  3888   …..   21936950640377856 = 6^{21}
  6174   …..   139488284660368896 = 518616^3
  9216   …..   692533995824480256 = 96^9
13122   …..   2846239010076427776 = 1417176^3
18000   …..   10077696000000000000 = 2160000^3
23958   …..   31628127098367714816 = 3162456^3
31104   …..   89853749822987698176 = 4478976^3
39546   …..   234791019246486283776 = 6169176^3
49392   …..   571344013968870998016 = 8297856^3
60750   …..   1307544150375000000000 = 10935000^3
73728   …..   2836619246897071128576 = 14155776^3

 
Here are the first few triples:

\{(3240, \; 7776, \; 8424), \; (1944, \; 2592, \; 3240) \}
\{(207360, \; 497664, \; 539136), \; (124416, \; 165888, \; 207360) \}
\{(2361960, \; 5668704, \; 6141096), \; (1417176, \; 1889568, \; 2361960) \}
\{(13271040, \; 31850496, \; 34504704), \; (7962624, \; 10616832, \; 13271040) \}
\{(50625000, \; 121500000, \; 131625000), \; (30375000, \; 40500000, \; 50625000) \}
\{(151165440, \; 362797056, \; 393030144), \; (90699264, \; 120932352, \; 151165440) \}
\{(381182760, \; 914838624, \; 991075176), \; (228709656, \; 304946208, \; 381182760) \}
\{(849346560, \; 2038431744, \; 2208301056), \; (509607936, \; 679477248, \; 849346560) \}
\{(1721868840, \; 4132485216, \; 4476858984), \; (1033121304, \; 1377495072, \; 1721868840) \}
\{(3240000000, \; 7776000000, \; 8424000000), \; (1944000000, \; 2592000000, \; 3240000000) \}
\{(5739857640, \; 13775658336, \; 14923629864), \; (3443914584, \; 4591886112, \; 5739857640) \}
\{(9674588160, \; 23219011584, \; 25153929216), \; (5804752896, \; 7739670528, \; 9674588160) \}
\{(15638861160, \; 37533266784, \; 40661039016), \; (9383316696, \; 12511088928, \; 15638861160) \}
\{(24395696640, \; 58549671936, \; 63428811264), \; (14637417984, \; 19516557312, \; 24395696640) \}
\{(36905625000, \; 88573500000, \; 95954625000), \; (22143375000, \; 29524500000, \; 36905625000) \}
\{(54358179840, \; 130459631616, \; 141331267584), \; (32614907904, \; 43486543872, \; 54358179840) \}

 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 

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About benvitalis

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

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