Prove that the list of the numbers with the property described below is finite.

I claim that you cannot find a **k-digit number** **(k ≥ 12)** with this property.

**N.B.** There are less than 50,000 which have your property (in base 10)

__CASE #1__ :

__1364 & 6143__ : The digits of 1,3,6,4 raised to the power of digits 6,1,4,3.

6143 is a permutation of 1364.

**(1^6) + (3^1) + (6^4) + (4^3) = 1364 **

__4316 & 6143__ :

**(4^6) + (3^1) + (1^4) + (6^3) = 4316**

**1364 and 4316 are permutations of 6143**

__4355 & 5435__ :

**(4^5) + (3^4) + (5^3) + (5^5) = 4355 **

__067236 & 326706__ :

**(3^0) + (2^6) + (6^7) + (7^2) + (0^3) + (6^6) = 326706 **

__Other examples__: 15630, 17463, 48625, 38650

__CASE #2 __: The digits & powers representing the digits one number give a permutation of that number.

__3435 & 5343__ :

**(5^5) + (3^3) + (4^4) + (3^3) = 3435**

__CASE #3__ :

**4096 = 4^6 + 0^9 **

The digits **4** and **0** are raised to powers **6** and **9**, and all these digits represent the digits of the number

Other examples,

**397612 = 3^2 + 9^1 + 7^6 + 6^7+ 1^9+ 2^3**

48625 = 4^5 + 8^2 + 6^6 + 2^8 + 5^4

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## About benvitalis

math grad - Interest: Number theory

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