Curious Properties of Multi-Power
relations
Abstract:-
Some of
the interesting and inherent properties associated with the specific relation
a2 + b2 + c2 = 2 d where a + b = c are
investigated.
Introduction:-
In number theory there are many relations under equal sums of
like powers and unlimited kinds of multi-power relations with different number
terms in each side and with different exponents. When we impose some conditions,
some of them are allowed and others are forbidden.
The sum of any number of squares can be shown to be equal to
the sum of any number of squares. This property is not found in any other
powers other than square where one or more restrictions disallow such formations.
The Fermat’s Last Theorem is one such discrimination. In this article, some of the interesting and
inherent properties associated with the specific relation
a2 + b2 + c2 = 2 d where a + b = c are
investigated.
Property-1
An interesting property
associated with the relation a2 + b2 + c2 = 2
d where a + b = c is d+ ab = (a+b)2
= c2. One can prove it with the help of algebra, but it has more
inherent properties. It is exemplified with few typical examples
12 + 122
+ 132 = 2 x 157 ; 157 +
1 x 12 = 169 = 132
22 + 112
+ 132 = 2 x 147 ; 147 +
2 x 11 = 169 = 132
32 + 102
+ 132 = 2 x 139 ; 139 +
3 x 10 = 169 = 132
42 + 92 + 132 = 2 x 133 ; 133 + 4
x 9 = 169 = 132
52 + 82
+ 132 = 2 x129 ;
129 + 5 x 8 = 169 = 132
62 + 72 + 132
= 2 x127 ; 127 + 6 x 7 = 169 = 132
Property-2
If a2 + b2
+ c2 = 2 d where a + b = c, then
a4 + b4
+ c4 = 2 d2
-2-
This is shown with some
numerical examples.
12 + 72
+ 82 = 2 x
57 and 14 + 74 + 84
= 6498 = 2 x 572
22 + 62 + 82 = 2 x
52 and 24 + 64 + 84 = 5408 = 2 x 522
32 + 52 + 82 =
2 x 49 and 34 + 54 + 84 = 4802 = 2 x 492
42 + 42 + 82 = 2 x
48 and 44 + 44
+ 84 = 4608 = 2 x 482
If d happens to be a square
number (d = e2) ,then the relation turns into a4 + b4
+ c4 = 2 e4. For example,
74 + 84 + 154 = 2 x 1692 =
2 x 134
54 + 164
+ 214 = 2 x 3612 = 2 x 194
94 + 154
+ 244 = 2 x 4412 = 2 x 214
114 + 244 + 354 = 2 x 9612
= 2 x 314
144 + 164
+ 304 = 2 x 6762 = 2 x 264
Combining these two
properties, we arrive yet another relation
a2 + b2
+ c2 = 2 d
Squaring both sides, (a2
+ b2 + c2)2 = (2d)2 = a4
+ b4 + c4 + 2(a2b2 + b2c2 + c2a2)
But a4 + b4
+ c4 = 2d2 which demands that (a2b2
+ b2c2 + c2a2)
= d2
Taking the case 14 + 74 + 84
= 2 x 572 , we
have
12 72 + 72 82 + 82 12
= 72 + 562 + 82 = 572
Property-3
If c = ab
and b = a+ 1 then a2 +
b2 + c2 = d2 where d is equal to c+1. For example,
12 + 22 + 22 = 32 ; 14 + 24 + 24 = 33 = 3 x 11
22 + 32 + 62 = 72 ; 24 + 34 + 64
= 1393 = 7 x199
32 + 42 + 122 = 132 ;34
+ 44 + 124 = 21073 = 13 x 1621
42 + 52 + 202 = 212 ; 44 + 54
+ 204 =160881 = 21 x 7661
It shows that a4 + b4
+ c4 invariably
has d as a factor
-3-
The readers may develop curiosity to
disclose many other properties associated with this type of conditional relations.
.................
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