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Topic: Sum Powers Of 2 And 5, Get Z (Read 700 times) |
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K Sengupta
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Sum Powers Of 2 And 5, Get Z
« on: Aug 6th, 2007, 8:15am » |
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Determine all possible non-negative integers (X, Y, Z) satisfying this equation:
2X + 5Y = 3Z
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Eigenray
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Re: Sum Powers Of 2 And 5, Get Z
« Reply #1 on: Aug 6th, 2007, 10:38am » |
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A start: note that z>0, and x>0 by parity. Suppose now that x>2 and y>0.
Mod 3, we see that x  y mod 2.
Mod 5, we see that x=z mod 2, either by enumerating the possibilities or because (-1)x=(2x|5) = (3z|5) = (-1)z.
Mod 8, we see that y=z=0 mod 2.
Together we get a contradiction. So either:
(1) y=0, which gives 2x + 1 = 3z, which gives (x,z) = (1,1) or (3,2). [If x>1, then z=2a is even, and 2x = (3a-1)(3a+1) forces a=1.]
(2) x < 3, and y>0.
If x=2, then 4 + 5y = 3z, and mod 4, we see z=2k is even, and then 5y = (3k+2)(3k-2). This is only possible if 3k-2=1, 3k+2=5, which gives the solution (x,y,z) = (2,1,2).
The remaining case is x=1, or
2 + 5y = 3z.
Mod 3, y=2k is even. This reduces to 5k + i 2 =  (1+i2)z when k is even, and 5k + i2 = (1-i2)z when k is odd.
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| « Last Edit: Aug 8th, 2007, 12:01am by Eigenray » |
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Eigenray
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Re: Sum Powers Of 2 And 5, Get Z
« Reply #2 on: Aug 7th, 2007, 7:12am » |
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Ad hoc computer-assisted solution:
We have 2 + 5y = 3z.
| hidden: | Suppose z>3. Working mod 34, we find that y = 20 mod  (34). Thus y = 20 + 54k. Now 554=1 mod 109, so 3z = 2+520 mod 109, which has no solutions. So z  3. |
It follows that the only solutions are (1,0,1), (3,0,2), (2,1,2), and (1,2,3).
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Hippo
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Re: Sum Powers Of 2 And 5, Get Z
« Reply #3 on: Aug 7th, 2007, 3:56pm » |
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on Aug 6th, 2007, 10:38am, Eigenray wrote:A start: note that z>0, and x>0 by parity. Suppose now that x>2 and y>0.
Mod 3, we see that x = -y mod 2.
Mod 5, we see that x=z mod 2, either by enumerating the possibilities or because (-1)x=(2x|5) = (3z|5) = (-1)z.
Mod 8, we see that y=z=0 mod 2.
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I get Mod 3: x+y=1 MOD 2.
Mod 5: x=z MOD 4.
Mod 8: y=z MOD 7.
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Eigenray
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Re: Sum Powers Of 2 And 5, Get Z
« Reply #4 on: Aug 8th, 2007, 12:04am » |
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on Aug 7th, 2007, 3:56pm, Hippo wrote:
You're right, what I had typed was not what I meant.
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That should be x=-z MOD 4; in particular x=z mod 2.
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I can't see how you're getting that. Mod 8, 5y = 1,5,1,5,..., while 3z = 1,3,1,3,.... So they are equal when y=z=0 mod 2.
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Hippo
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Re: Sum Powers Of 2 And 5, Get Z
« Reply #5 on: Aug 9th, 2007, 5:46am » |
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Oops ... you are right ... I had find a mistake and made two others ... sorry once again ... I didn't spent a lot of time with the answer and it is not what I usually do.
 Mod 5: It should be x+z=2 mod 4 ... ok ... .
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| « Last Edit: Aug 9th, 2007, 6:34am by Hippo » |
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Eigenray
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Re: Sum Powers Of 2 And 5, Get Z
« Reply #6 on: Aug 9th, 2007, 5:59am » |
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on Aug 9th, 2007, 5:46am, Hippo wrote:| Mod 5: It should be x+z=2 mod 4 |
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Mod 5, 3 = 2-1, so 2x = 3z = 2-z implies x=-z mod 4.
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Hippo
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Re: Sum Powers Of 2 And 5, Get Z
« Reply #7 on: Aug 9th, 2007, 6:28am » |
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Ooops, oops I had solved 2x+3z=0 now. I should not write more about this topic 
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srn437
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Re: Sum Powers Of 2 And 5, Get Z
« Reply #8 on: Aug 26th, 2007, 9:05pm » |
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x, y, and z all equal infinity(positive or negative).
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Sameer
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Re: Sum Powers Of 2 And 5, Get Z
« Reply #9 on: Aug 26th, 2007, 9:12pm » |
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on Aug 26th, 2007, 9:05pm, srn347 wrote:| x, y, and z all equal infinity(positive or negative). |
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Infinity is not a number an integer
Edit: As pointed out by pex!! 
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| « Last Edit: Aug 27th, 2007, 8:31pm by Sameer » |
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"Obvious" is the most dangerous word in mathematics.
--Bell, Eric Temple
Proof is an idol before which the mathematician tortures himself.
Sir Arthur Eddington, quoted in Bridges to Infinity
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mikedagr8
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Re: Sum Powers Of 2 And 5, Get Z
« Reply #10 on: Aug 26th, 2007, 9:45pm » |
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Hahahaha, good call, you beat me to it. Infinity means, well, just wiktionary it, I could explain it in my own terms, except wiki does it better than me.
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| « Last Edit: Aug 26th, 2007, 9:46pm by mikedagr8 » |
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"It's not that I'm correct, it's that you're just not correct, and so; I am right." - M.P.E.
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pex
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Re: Sum Powers Of 2 And 5, Get Z
« Reply #11 on: Aug 26th, 2007, 11:37pm » |
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on Aug 26th, 2007, 9:12pm, Sameer wrote:
Infinity is not a number. |
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Of course it is - just not a real number, and most clearly not an integer, as this puzzle requires.
See Affinely Extended Real Numbers, Projectively Extended Real Numbers, among others.
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srn437
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Re: Sum Powers Of 2 And 5, Get Z
« Reply #12 on: Aug 27th, 2007, 10:23am » |
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Infinity is hyperreal. And how about x=1 y=2 z=3
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| « Last Edit: Sep 1st, 2007, 11:22pm by srn437 » |
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