a^k + 2b^k + 2c^k = d^k, where k = 4, 5

 

                                     (1)   a^4   +   2 b^4   +   2 c^4   =   d^4
                                     (2)   a^5   +   2 b^5   +   2 c^5   =   d^5

 
(1)

2^4   +   2^4   +   3^4   +   4^4   +   4^4   =   625   =   5^4

In other words,

[2, 2, 3, 4, 4]   =   [5]
 

Note that   3^2 + 4^2 = 5^2,     (3, 4, 5)   is a primitive Pythagorean triple

Find other Pythagorean-like solutions.

that is to say,   find   (a, b, c, d)   so that

a^4   +   2 b^4   +   2 c^4   =   d^4
and
b^2   +   c^2   =   d^2     or     a^2   +   b^2   =   d^2

 
Pythagorean triple
http://en.wikipedia.org/wiki/Pythagorean_triple

 

[4, 4, 6, 8, 8]   =   [10]
2*(4^4)   +   6^4   +   2*(8^4) = 10^4
[6, 8, 10]   is a multiple of   [3, 4, 5]

[6, 6, 9, 12, 12]   =   [15]
2*(6^4)   +   9^4   +   2*(12^4)   =   15^4
[9, 12, 15]   is a multiple of   [3, 4, 5]

[8, 8, 12, 16, 16]   =   [20]
2*(8^4)   +   12^4   +   2*(16^4)   =  20^4
[12, 16, 20]   is a multiple of   [3, 4, 5]

[10, 10, 15, 20, 20]   =   [25]
2*(10^4)   +   15^4   +   2*(20^4)   =   25^4
[15, 20, 25]   is a multiple of   [3, 4, 5]

[12, 12, 18, 24, 24]   =   [30]
2*(12^4)   +   18^4   +   2*(24^4)   =   30^4
[18, 24, 30]   is a multiple of   [3, 4, 5]

[14, 14, 21, 28, 28]   =   [35]
2*(14^4)   +   21^4   +   2*(28^4)   =   35^4
[21, 28, 35]   is a multiple of   [3, 4, 5]

[16, 16, 24, 32, 32]   =   [40]
2*(16^4)   +   24^4   +   2*(32^4)   =   40^4
[24, 32, 40]   is a multiple of   [3, 4, 5]

[18, 18, 27, 36, 36]   =   [45]
2*(18^4)   +   27^4   +   2*(36^4)   =   45^4
[27, 36, 45]   is a multiple of   [3, 4, 5]
 

and so on.

Can you find another primitive solution, that is, a solution not a multiple of   [3, 4, 5]   ?

 
 

(2)

526^5   +   526^5   +   1349^5   +   1349^5   +   1355^5   =   1685^5    
( = 13583119642053125)

Can you find other solutions?

 

                                                 ——————————————
 
There are 16 primitive Pythagorean triples with c ≤ 100 :

(3, 4, 5)      (5, 12, 13)      (8, 15, 17)      (7, 24, 25)
(20, 21, 29)      (12, 35, 37)      (9, 40, 41)      (28, 45, 53)
(11, 60, 61)      (16, 63, 65)      (33, 56, 65)      (48, 55, 73)
(13, 84, 85)      (36, 77, 85)      (39, 80, 89)      (65, 72, 97)

[2, 2, 2, 2, 2, 2, 2, 2, 8, 8, 8, 8, 8, 8, 8, 8, 15]   =   [17]
8*(2^4)   +   8*(8^4)   +   15^4   =   17^4

 
 
 

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About benvitalis

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

1 Response to a^k + 2b^k + 2c^k = d^k, where k = 4, 5

  1. David says:

    #!/usr/bin/python3
    # David
    # a^k + 2b^k + 2c^k = d^k, where k = 4, 5
    # (1) a^4 + 2 b^4 + 2 c^4 = d^4
    # (2) a^5 + 2 b^5 + 2 c^5 = d^5

    import math

    MAX = 2
    for k in range(4,6):
    for a in range(1,10**MAX):
    for b in range(1,10**MAX):
    for c in range(1,10**MAX):
    for d in range(1,10**MAX):
    if (
    (a**k)+(2*(b**k))+(2*(c**k)) == d**k and
    (math.hypot(a,b) or math.hypot(b,c)) == d
    ):
    print(‘Is ({0},{1},{3}) a Pythagorean-like triple? => {0}^{4} + 2*({1}^{4}) + 2*({2}^{4}) = {3}^{4}’.format(a,b,c,d,k))

    # some answers re (1) ….

    Is (3,4,5) a Pythagorean-like triple? => 3^4 + 2*(4^4) + 2*(2^4) = 5^4
    Is (6,8,10) a Pythagorean-like triple? => 6^4 + 2*(8^4) + 2*(4^4) = 10^4
    Is (9,12,15) a Pythagorean-like triple? => 9^4 + 2*(12^4) + 2*(6^4) = 15^4
    Is (12,16,20) a Pythagorean-like triple? => 12^4 + 2*(16^4) + 2*(8^4) = 20^4
    Is (15,20,25) a Pythagorean-like triple? => 15^4 + 2*(20^4) + 2*(10^4) = 25^4
    Is (18,24,30) a Pythagorean-like triple? => 18^4 + 2*(24^4) + 2*(12^4) = 30^4
    Is (21,28,35) a Pythagorean-like triple? => 21^4 + 2*(28^4) + 2*(14^4) = 35^4
    Is (24,32,40) a Pythagorean-like triple? => 24^4 + 2*(32^4) + 2*(16^4) = 40^4
    Is (27,36,45) a Pythagorean-like triple? => 27^4 + 2*(36^4) + 2*(18^4) = 45^4
    Is (30,40,50) a Pythagorean-like triple? => 30^4 + 2*(40^4) + 2*(20^4) = 50^4
    Is (33,44,55) a Pythagorean-like triple? => 33^4 + 2*(44^4) + 2*(22^4) = 55^4
    Is (36,48,60) a Pythagorean-like triple? => 36^4 + 2*(48^4) + 2*(24^4) = 60^4
    Is (39,52,65) a Pythagorean-like triple? => 39^4 + 2*(52^4) + 2*(26^4) = 65^4
    Is (42,56,70) a Pythagorean-like triple? => 42^4 + 2*(56^4) + 2*(28^4) = 70^4
    Is (45,60,75) a Pythagorean-like triple? => 45^4 + 2*(60^4) + 2*(30^4) = 75^4
    Is (48,64,80) a Pythagorean-like triple? => 48^4 + 2*(64^4) + 2*(32^4) = 80^4
    Is (51,68,85) a Pythagorean-like triple? => 51^4 + 2*(68^4) + 2*(34^4) = 85^4
    Is (54,72,90) a Pythagorean-like triple? => 54^4 + 2*(72^4) + 2*(36^4) = 90^4
    Is (57,76,95) a Pythagorean-like triple? => 57^4 + 2*(76^4) + 2*(38^4) = 95^

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