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Topic: Sum The Exponents, Get Perfect Power? (Read 366 times) |
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K Sengupta
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Sum The Exponents, Get Perfect Power?
« on: Jul 20th, 2007, 11:28am » |
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Analytically determine, whether or not there exist any positive prime number x such that :
2x + 5x is a perfect power.
NOTE:
The definition of perfect power is given in:
http://mathworld.wolfram.com/PerfectPower.html
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FiBsTeR
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Re: Sum The Exponents, Get Perfect Power?
« Reply #1 on: Jul 20th, 2007, 1:04pm » |
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(silly post removed)
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| « Last Edit: Jul 20th, 2007, 1:35pm by FiBsTeR » |
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SMQ
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Re: Sum The Exponents, Get Perfect Power?
« Reply #2 on: Jul 20th, 2007, 1:30pm » |
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Come again?
(pointing out of silly mistake removed -- no worries, mate, we all do it from time to time...)
--SMQ
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| « Last Edit: Jul 20th, 2007, 1:51pm by SMQ » |
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--SMQ
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FiBsTeR
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Re: Sum The Exponents, Get Perfect Power?
« Reply #3 on: Jul 20th, 2007, 1:36pm » |
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Oops.
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SMQ
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Re: Sum The Exponents, Get Perfect Power?
« Reply #4 on: Jul 23rd, 2007, 2:29pm » |
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7 | 2x + 5x for all odd x, but 49 | 2x + 5x iff x is congruent to 7, 21 or 35 mod 42. Thus, for any odd prime x other than 7, 2x + 5x is multiple of 7 but not 49, and therefore not a perfect power.
22 + 52 = 29 is not a perfect power. 27 + 57 = 78253 is not a perfect power.
Therefore there is no x for which 2x + 5x is a perfect power.
--SMQ
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K Sengupta
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Re: Sum The Exponents, Get Perfect Power?
« Reply #5 on: Jul 24th, 2007, 1:17am » |
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on Jul 23rd, 2007, 2:29pm, SMQ wrote:7 | 2x + 5x for all odd x, but 49 | 2x + 5x iff x is congruent to 7, 21 or 35 mod 42. Thus, for any odd prime x other than 7, 2x + 5x is multiple of 7 but not 49, and therefore not a perfect power.
22 + 52 = 29 is not a perfect power. 27 + 57 = 78253 is not a perfect power.
Therefore there is no x for which 2x + 5x is a perfect power.
--SMQ |
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Well done, SMQ.
This was precisely the solution I was looking for.
However, your proof that :
(2^x+5^x)/7 = 0 (mod 7) iff x = 0 (Mod 7)
varies very slightly from my own methodology.
My proposed solution to the given problem is furnished hereunder as follows:
For x =2; 2^x + 5^x = 29, which is not a perfect power.
For x>=3, x must be odd, since x is prime.
Thus, (2^x + 5^x) is divisible by 7 for all x>=3, and accordingly, the perfect power, if it exists, must be a power of 7.
Hence substituting x = 2y+1, for positive integer y, we observe that:
(2^x + 5^x)/7
= (2^2y 2^(2y-1)*5 + 2^(2y-2)*5^2 2^(2y-3)*(5^3)
..- 2* 5^(2y-1) + 5^(2y))
= {2^(2y) + 2^(2y-1)*2 +2^(2y-2)*2^2 +
+ 2*2^(2y-1) + 2(2y)}(Mod 7)
= ((2y+1)*2^(2y) (Mod 7)
= x* 2^(x-1)(Mod 7)
Since (2, 7) = 1, it now follows that (2^x + 5^x)/7 = 0 (Mod 7) iff x = 7t for some integer t.
However, we know that x is prime and accordingly, x= 7
Therefore, 2^x + 5^x may correspond to a perfect power only when x=7.
However, in reality, we observe that 2^7 + 5^7 = (7^2)*(19)*(83), and consequently:
2^x + 5^x can never correspond to a perfect power for prime x.
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