# Fourth powers

There are only three numbers that can be written as the sum of fourth powers of their digits:

Find the smallest number that can be written as the sum of fifth powers of its digits.

Reveal Solution
It helps to know the fifth powers of each possible digit so here they are:

 0 0 5 3125 1 1 6 7776 2 32 7 16807 3 243 8 32768 4 1024 9 59049

There are a few simple conclusions that can be made from this:

• There are no one digit numbers (as we don’t include or ).
• Two digit numbers can only use , , since is a three digit number and therefore there are none of these.
• Three digit numbers would have to contain at least one since would only give a total of and therefore there are none of these.

For four digit numbers , , , , , , can be used

The first digit is and you can’t have s or s since both give totals . One of the digits must be and none of these combinations work.

The first digit is and you can’t have s or s since both give totals . One of the digits must be and none of these combinations work.

The first digit is and you can’t have s since this gives totals . One of the digits must be or a and none of these combinations work.

The first digit is and you can’t have s since this gives totals . One of the other digits must be or a .

None of the combinations with a second work.

Examining numbers of the form _ _ _, with one gives:

The next smallest solution is:

The following short Python script can be used to show that these are the only 4-digit solutions:

def conway(n):
for a in range(0,10):
for b in range(0,10):
for c in range(0,10):
for d in range(0,10):
m=1000*a+100*b+10*c+d
if a**n+b**n+c**n+d**n==m and m>1:
print(m)

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