Decimal integers with nonzero digits and sum of powers puzzle

K Sengupta

Determine all possible positive decimal integer(s) P = X1X2X3….Xn, where P>=2 with none of the digits in P being zero, that satisfy this equation:

P = X1^X1 + X2^X2 + ……+ Xn^Xn

(For example, P = 234 cannot be a solution since 2^2 + 3^3 + 4^4 is equal to 287, not 234.)

Notes:

(i) X1X2X3….Xn denotes the concatenation of the digits X1, X2, …, Xn and do not represent the product of the digits.

(ii) P cannot admit any leading zero.

davee123

I'm assuming that X1, X2, X3 are all the individual *digits* of P?

DaveE

K Sengupta

Yes, each of X1,X2, ....,Xn correspond to an individual digit of P.

skeptic2

Can any digit be the same as another digit?

Gokul43201

...or presumably, any other zeros.

jimmysnyder

Here are a few:

Of these only the first meets the unstated condition that all digits be distinct. The third one fails to meet the unstated condition that no digits be 0. I used a brute force method to get this partial solution. I do not know if this list is complete.

davee123

That also assumes that 0^0 = 0-- I thought the more accepted solution was that 0^0 = 1?

Otherwise, I think there's an upper bound of roughly 3.4 billion. Beyond that I think the rate at which the sum of the powers increases is capped (since 9^9 is the highest sum a digit can contribute), and the number itself is increasing more quickly.

DaveE

jimmysnyder

Good catch. There was a bug in my code. So I have only found 2 solutions.