## a^2 + b^2 + c^2 = d^2 + e^2 + f^2, and a*b*c=d*e*f — Part 2

If

$a^2 \; + \; b^2 \; + \; c^2 \; = \; d^2 \; + \; e^2 \; + \; f^2$
$a \cdot b \cdot c \; = \; d \cdot e \cdot f$

then,

$(a+b+c) + (a-b-c) + (-a-b+c) + (-a+b-c)$
$= (d+e+f) + (d-e-f) + (-d-e+f) + (-d+e-f)$

$(a+b+c)^2 + (a-b-c)^2 + (-a-b+c)^2 + (-a+b-c)^2$
$= (d+e+f)^2 + (d-e-f)^2 + (-d-e+f)^2 + (-d+e-f)^2$

$(a+b+c)^3 + (a-b-c)^3 + (-a-b+c)^3 + (-a+b-c)^3$
$= (d+e+f)^3 + (d-e-f)^3 + (-d-e+f)^3 + (-d+e-f)^3$

$(a+b+c)^5 + (a-b-c)^5 + (-a-b+c)^5 + (-a+b-c)^5$
$= (d+e+f)^5 + (d-e-f)^5 + (-d-e+f)^5 + (-d+e-f)^5$

$a^2 \; + \; b^2 \; + \; c^2 \; = \; d^2 \; + \; e^2 \; + \; f^2 \; = \; S$

$a \cdot b \cdot c \; = \; d \cdot e \cdot f \; = \; P$

where   a = 2

#1

$2^2 + 11^2 + 20^2 = 4^2 + 5^2 + 22^2 = 525$
$2 \; \times \; 11 \; \times \; 20 \; = \; 4 \; \times \; 5 \; \times \; 22 \; = \; 440$

$33 + (-29) + 7 + (-11) = 31 + (-23) + 13 + (-21) = 0$
$33^2 + (-29)^2 + 7^2 + (-11)^2 = 31^2 + (-23)^2 + 13^2 + (-21)^2 = 2100$
$33^3 + (-29)^3 + 7^3 + (-11)^3 = 31^3 + (-23)^3 + 13^3 + (-21)^3 = 10560$
$33^5 + (-29)^5 + 7^5 + (-11)^5 = 31^5 + (-23)^5 + 13^5 + (-21)^5 = 18480000$

#2

$2^2 + 13^2 + 24^2 = 3^2 + 8^2 + 26^2 = 749$
$2 \; \times \; 13 \; \times \; 24 \; = \; 3 \; \times \; 8 \; \times \; 26 \; = \; 624$

$39 + (-35) + 9 + (-13) = 37 + (-31) + 15 + (-21) = 0$
$39^2 + (-35)^2 + 9^2 + (-13)^2 = 37^2 + (-31)^2 + 15^2 + (-21)^2 = 2996$
$39^3 + (-35)^3 + 9^3 + (-13)^3 = 37^3 + (-31)^3 + 15^3 + (-21)^3 = 14976$
$39^5 + (-35)^5 + 9^5 + (-13)^5 = 37^5 + (-31)^5 + 15^5 + (-21)^5 = 37390080$

#3

$2^2 + 20^2 + 21^2 = 5^2 + 6^2 + 28^2 = 845$
$2 \; \times \; 20 \; \times \; 21 \; = \; 5 \; \times \; 6 \; \times \; 28 \; = \; 840$

$43 + (-39) + (-1) + (-3) = 39 + (-29) + 17 + (-27) = 0$
$43^2 + (-39)^2 + (-1)^2 + (-3)^2 = 39^2 + (-29)^2 + 17^2 + (-27)^2 = 3380$
$43^3 + (-39)^3 + (-1)^3 + (-3)^3 = 39^3 + (-29)^3 + 17^3 + (-27)^3 = 20160$
$43^5 + (-39)^5 + (-1)^5 + (-3)^5 = 39^5 + (-29)^5 + 17^5 + (-27)^5 = 56784000$

#4

$2^2 + 20^2 + 24^2 = 4^2 + 8^2 + 30^2 = 980$
$2 \; \times \; 20 \; \times \; 24 \; = \; 4 \; \times \; 8 \; \times \; 30 \; = \; 960$

$46 + (-42) + 2 + (-6) = 42 + (-34) + 18 + (-26) = 0$
$46^2 + (-42)^2 + 2^2 + (-6)^2 = 42^2 + (-34)^2 + 18^2 + (-26)^2 = 3920$
$46^3 + (-42)^3 + 2^3 + (-6)^3 = 42^3 + (-34)^3 + 18^3 + (-26)^3 = 23040$
$46^5 + (-42)^5 + 2^5 + (-6)^5 = 42^5 + (-34)^5 + 18^5 + (-26)^5 = 75264000$

#5

$2^2 + 24^2 + 35^2 = 3^2 + 14^2 + 40^2 = 1805$
$2 \; \times \; 24 \; \times \; 35 \; = \; 3 \; \times \; 14 \; \times \; 40 \; = \; 1680$

$61 + (-57) + 9 + (-13) = 57 + (-51) + 23 + (-29) = 0$
$61^2 + (-57)^2 + 9^2 + (-13)^2 = 57^2 + (-51)^2 + 23^2 + (-29)^2 = 7220$
$61^3 + (-57)^3 + 9^3 + (-13)^3 = 57^3 + (-51)^3 + 23^3 + (-29)^3 = 40320$
$61^5 + (-57)^5 + 9^5 + (-13)^5 = 57^5 + (-51)^5 + 23^5 + (-29)^5 = 242592000$

#6

$2^2 + 29^2 + 52^2 = 4^2 + 13^2 + 58^2 = 3549$
$2 \; \times \; 29 \; \times \; 52 \; = \; 4 \; \times \; 13 \; \times \; 58 \; = \; 3016$

$83 + (-79) + 21 + (-25) = 75 + (-67) + 41 + (-49) = 0$
$83^2 + (-79)^2 + 21^2 + (-25)^2 = 75^2 + (-67)^2 + 41^2 + (-49)^2 = 14196$
$83^3 + (-79)^3 + 21^3 + (-25)^3 = 75^3 + (-67)^3 + 41^3 + (-49)^3 = 72384$
$83^5 + (-79)^5 + 21^5 + (-25)^5 = 75^5 + (-67)^5 + 41^5 + (-49)^5 = 856302720$

#7

$2^2 + 31^2 + 55^2 = 5^2 + 11^2 + 62^2 = 3990$
$2 \; \times \; 31 \; \times \; 55 \; = \; 5 \; \times \; 11 \; \times \; 62 \; = \; 3410$

$88 + (-84) + 22 + (-26) = 78 + (-68) + 46 + (-56) = 0$
$88^2 + (-84)^2 + 22^2 + (-26)^2 = 78^2 + (-68)^2 + 46^2 + (-56)^2 = 15960$
$88^3 + (-84)^3 + 22^3 + (-26)^3 = 78^3 + (-68)^3 + 46^3 + (-56)^3 = 81840$
$88^5 + (-84)^5 + 22^5 + (-26)^5 = 78^5 + (-68)^5 + 46^5 + (-56)^5 = 1088472000$

#8

$2^2 + 31^2 + 57^2 = 3^2 + 19^2 + 62^2 = 4214$
$2 \; \times \; 31 \; \times \; 57 \; = \; 3 \; \times \; 19 \; \times \; 62 \; = \; 3534$

$90 + (-86) + 24 + (-28) = 84 + (-78) + 40 + (-46) = 0$
$90^2 + (-86)^2 + 24^2 + (-28)^2 = 84^2 + (-78)^2 + 40^2 + (-46)^2 = 16856$
$90^3 + (-86)^3 + 24^3 + (-28)^3 = 84^3 + (-78)^3 + 40^3 + (-46)^3 = 84816$
$90^5 + (-86)^5 + 24^5 + (-28)^5 = 84^5 + (-78)^5 + 40^5 + (-46)^5 = 1191382080$

#9

$2^2 + 34^2 + 60^2 = 6^2 + 10^2 + 68^2 = 4760$
$2 \; \times \; 34 \; \times \; 60 \; = \; 6 \; \times \; 10 \; \times \; 68 \; = \; 4080$

$96 + (-92) + 24 + (-28) = 84 + (-72) + 52 + (-64) = 0$
$96^2 + (-92)^2 + 24^2 + (-28)^2 = 84^2 + (-72)^2 + 52^2 + (-64)^2 = 19040$
$96^3 + (-92)^3 + 24^3 + (-28)^3 = 84^3 + (-72)^3 + 52^3 + (-64)^3 = 97920$
$96^5 + (-92)^5 + 24^5 + (-28)^5 = 84^5 + (-72)^5 + 52^5 + (-64)^5 = 1553664000$

#10

$2^2 + 36^2 + 39^2 = 6^2 + 9^2 + 52^2 = 2821$
$2 \; \times \; 36 \; \times \; 39 \; = \; 6 \; \times \; 9 \; \times \; 52 \; = \; 2808$

$77 + (-73) + 1 + (-5) = 67 + (-55) + 37 + (-49) = 0$
$77^2 + (-73)^2 + 1^2 + (-5)^2 = 67^2 + (-55)^2 + 37^2 + (-49)^2 = 11284$
$77^3 + (-73)^3 + 1^3 + (-5)^3 = 67^3 + (-55)^3 + 37^3 + (-49)^3 = 67392$
$77^5 + (-73)^5 + 1^5 + (-5)^5 = 67^5 + (-55)^5 + 37^5 + (-49)^5 = 633709440$

#11

$2^2 + 36^2 + 45^2 = 3^2 + 20^2 + 54^2 = 3325$
$2 \; \times \; 36 \; \times \; 45 \; = \; 3 \; \times \; 20 \; \times \; 54 \; = \; 3240$

$83 + (-79) + 7 + (-11) = 77 + (-71) + 31 + (-37) = 0$
$83^2 + (-79)^2 + 7^2 + (-11)^2 = 77^2 + (-71)^2 + 31^2 + (-37)^2 = 13300$
$83^3 + (-79)^3 + 7^3 + (-11)^3 = 77^3 + (-71)^3 + 31^3 + (-37)^3 = 77760$
$83^5 + (-79)^5 + 7^5 + (-11)^5 = 77^5 + (-71)^5 + 31^5 + (-37)^5 = 861840000$

#12

$2^2 + 39^2 + 55^2 = 5^2 + 13^2 + 66^2 = 4550$
$2 \; \times \; 39 \; \times \; 55 \; = \; 5 \; \times \; 13 \; \times \; 66 \; = \; 4290$

$96 + (-92) + 14 + (-18) = 84 + (-74) + 48 + (-58) = 0$
$96^2 + (-92)^2 + 14^2 + (-18)^2 = 84^2 + (-74)^2 + 48^2 + (-58)^2 = 18200$
$96^3 + (-92)^3 + 14^3 + (-18)^3 = 84^3 + (-74)^3 + 48^3 + (-58)^3 = 102960$
$96^5 + (-92)^5 + 14^5 + (-18)^5 = 84^5 + (-74)^5 + 48^5 + (-58)^5 = 1561560000$

#13

$2^2 + 41^2 + 72^2 = 8^2 + 9^2 + 82^2 = 6869$
$2 \; \times \; 41 \; \times \; 72 \; = \; 8 \; \times \; 9 \; \times \; 82 \; = \; 5904$

$115 + (-111) + 29 + (-33) = 99 + (-83) + 65 + (-81) = 0$
$115^2 + (-111)^2 + 29^2 + (-33)^2 = 99^2 + (-83)^2 + 65^2 + (-81)^2 = 27476$
$115^3 + (-111)^3 + 29^3 + (-33)^3 = 99^3 + (-83)^3 + 65^3 + (-81)^3 = 141696$
$115^5 + (-111)^5 + 29^5 + (-33)^5 = 99^5 + (-83)^5 + 65^5 + (-81)^5 = 3244366080$

#14

$2^2 + 51^2 + 52^2 = 3^2 + 26^2 + 68^2 = 5309$
$2 \; \times \; 51 \; \times \; 52 \; = \; 3 \; \times \; 26 \; \times \; 68 \; = \; 5304$

$105 + (-101) + (-1) + (-3) = 97 + (-91) + 39 + (-45) = 0$
$105^2 + (-101)^2 + (-1)^2 + (-3)^2 = 97^2 + (-91)^2 + 39^2 + (-45)^2 = 21236$
$105^3 + (-101)^3 + (-1)^3 + (-3)^3 = 97^3 + (-91)^3 + 39^3 + (-45)^3 = 127296$
$105^5 + (-101)^5 + (-1)^5 + (-3)^5 = 97^5 + (-91)^5 + 39^5 + (-45)^5 = 2252714880$

#15

$2^2 + 56^2 + 84^2 = 4^2 + 24^2 + 98^2 = 10196$
$2 \; \times \; 56 \; \times \; 84 \; = \; 4 \; \times \; 24 \; \times \; 98 \; = \; 9408$

$142 + (-138) + 26 + (-30) = 126 + (-118) + 70 + (-78) = 0$
$142^2 + (-138)^2 + 26^2 + (-30)^2 = 126^2 + (-118)^2 + 70^2 + (-78)^2 = 40784$
$142^3 + (-138)^3 + 26^3 + (-30)^3 = 126^3 + (-118)^3 + 70^3 + (-78)^3 = 225792$
$142^5 + (-138)^5 + 26^5 + (-30)^5 = 126^5 + (-118)^5 + 70^5 + (-78)^5 = 7673917440$

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## About benvitalis

math grad - Interest: Number theory
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