__Curious Properties of Multi-Power relations__

__Abstract:-__**Some of the interesting and inherent properties associated with the specific relation**

**a**

^{2 }+ b^{2}+ c^{2 }= 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**

**a**

^{2 }+ b^{2}+ c^{2 }= 2 d where a + b = c are investigated.

__Property-1__**An interesting property associated with the relation a**

^{2 }+ b^{2}+ c^{2 }= 2 d where a + b = c is d+ ab = (a+b)^{2}= c^{2}. One can prove it with the help of algebra, but it has more inherent properties. It is exemplified with few typical examples**1**

^{2}+ 12^{2 }+ 13^{2 }= 2 x 157 ; 157 + 1 x 12 = 169 = 13^{2}**2**

^{2}+ 11^{2 }+ 13^{2 }= 2 x 147 ; 147 + 2 x 11 = 169 = 13^{2}**3**

^{2}+ 10^{2 }+ 13^{2 }= 2 x 139 ; 139 + 3 x 10 = 169 = 13^{2}**4**

^{2 }+ 9^{2 }+ 13^{2 }= 2 x 133 ; 133 + 4 x 9 = 169 = 13^{2}**5**

^{2}+ 8^{2 }+ 13^{2 }= 2 x129 ; 129 + 5 x 8 = 169 = 13^{2}**6**

^{2 }+ 7^{2 }+ 13^{2 }= 2 x127 ; 127 + 6 x 7 = 169 = 13^{2 }

__Property-2__**If a**

^{2 }+ b^{2}+ c^{2 }= 2 d where a + b = c, then**a**

^{4 }+ b^{4 }+ c^{4 }= 2 d^{2 }**-2-**

**This is shown with some numerical examples.**

**1**

^{2 }+ 7^{2 }+ 8^{2 }= 2 x 57 and 1^{4 }+ 7^{4 }+ 8^{4 }= 6498 = 2 x 57^{2}**2**

^{2 }+ 6^{2 }+ 8^{2 }= 2 x 52 and 2^{4 }+ 6^{4 }+ 8^{4 }= 5408 = 2 x 52^{2 }**3**

^{2 }+ 5^{2 }+ 8^{2 }= 2 x 49 and 3^{4 }+ 5^{4 }+ 8^{4 }= 4802 = 2 x 49^{2 }^{}**4**

^{2 }+ 4^{2 }+ 8^{2 }= 2 x 48 and 4^{4 }+ 4^{4 }+ 8^{4 }= 4608 = 2 x 48^{2}**If d happens to be a square number (d = e**

^{2}) ,then the relation turns into a^{4 }+ b^{4 }+ c^{4 }= 2 e^{4. }For example,**7**

^{4 }+ 8^{4 }+ 15^{4 }= 2 x 169^{2 }=^{ }2 x 13^{4}**5**

^{4 }+ 16^{4 }+ 21^{4 }= 2 x 361^{2}= 2 x 19^{4}**9**

^{4}+ 15^{4 }+ 24^{4 }= 2 x 441^{2 }= 2 x 21^{4}**11**

^{4 }+ 24^{4 }+ 35^{4 }= 2 x 961^{2 }= 2 x 31^{4}**14**

^{4 }+ 16^{4 }+ 30^{4 }= 2 x 676^{2 }= 2 x 26^{4}^{}**Combining these two properties, we arrive yet another relation**

**a**

^{2 }+ b^{2}+ c^{2 }= 2 d**Squaring both sides, (a**

^{2 }+ b^{2}+ c^{2})^{2 }= (2d)^{2 }= a^{4 }+ b^{4 }+ c^{4 }+ 2(a^{2}b^{2 }+ b^{2}c^{2}+ c^{2}a^{2})**But a**

^{4 }+ b^{4 }+ c^{4}= 2d^{2 }which demands that^{ }(a^{2}b^{2 }+ b^{2}c^{2}+ c^{2}a^{2}) = d^{2}**Taking the case 1**

^{4 }+ 7^{4 }+ 8^{4 }= 2 x 57^{2 }, we have**1**

^{2 }7^{2 }+ 7^{2 }8^{2 }+ 8^{2 }1^{2 }= 7^{2}+ 56^{2 }+ 8^{2 }=^{ }57^{2}

__Property-3__**If c = ab and b = a+ 1 then a**

^{2 }+ b^{2 }+ c^{2 }= d^{2 }where d is equal to c+1. For example,**1**

^{2 }+ 2^{2 }+ 2^{2 }= 3^{2 }; 1^{4 }+ 2^{4 }+ 2^{4 }= 33 = 3 x 11**2**

^{2 }+ 3^{2}+ 6^{2 }= 7^{2 }; 2^{4 }+ 3^{4 }+ 6^{4 }= 1393 = 7 x199^{}**3**

^{2}+ 4^{2 }+ 12^{2 }= 13^{2};3^{4}+ 4^{4 }+ 12^{4 }= 21073 = 13 x 1621^{}**4**

^{2}+ 5^{2 }+ 20^{2 }= 21^{2 }; 4^{4}+ 5^{4}+ 20^{4}=160881 = 21 x 7661**It shows that a**

^{4 }+ b^{4 }+ c^{4}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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