Sum of two fifth powers!

Number Theory Level 4

Find the nine-digit number of the form xxxxyyyyy that can be written as a sum of fifth powers of two positive integers.

Proposed by Titu Andreescu,University of Texas at Dallas, USA

The answer is 777700000.

1 solution

Rajen Kapur
Feb 18, 2015

The given number xxxxyyyyy can be viewed as parts 11 * 101 * 10^5x and 41 * 271y, that might have a common factor as their sum equals a^5 + b^5 with (a+b) being a factor (a and b as positive numbers). After due attention to finding a common factor, it may be concluded that y = 0. Then on the problem reduces to finding two integers whose whose fifth powers add up to xxxx. As 6^5 = 7776, 60 and 10 are the two positive numbers whose fifth powers add up to 777700000.

Can you explain why must they have a common factor in the sum of two positive integers?

Note: When using multiple *, you have to place spaces between them, otherwise the markdown code will start to convert them into italics or bold font.

I've edited your solution for your reference.

Calvin Lin Staff - 6 years, 3 months ago

a^5 + b^5 has a factor a + b.

Rajen Kapur - 6 years, 3 months ago

Can you add that into your solution?

I also do not see how $a + b | nx + y b$ would imply that $\gcd(x, y) \neq 1$ .

Calvin Lin Staff - 6 years, 3 months ago

Could you please elaborate "The given number can be viewed as x111012^55^5 + y41271"?? I am not able to understand the notation, I think?!

Vishnu H Nair - 6 years, 3 months ago

x and y are two digits. 1111 = 11 x 101 and 11111 = 41 x 271 are simple factors. 2^5 and 5^5 multiply out to 100000.

Rajen Kapur - 6 years, 3 months ago

Sum of two fifth powers!

Proposed by Titu Andreescu,University of Texas at Dallas, USA

The answer is 777700000.

1 solution

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