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Topic: Prime Time (Read 231 times) |
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ThudnBlunder
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For which positive integer values of n is n4 + 4n prime?
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TenaliRaman
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Re: Prime Time
« Reply #1 on: Nov 29th, 2003, 6:38am » |
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(not a complete solution)
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obviously n cannot be even bcos for n even , the given expression is definitely composite.
so we have to look at odds.
note that for n odd, 4n has last digit as 4
now for n's ending with 1,3,7,9 the n4 ends with a 1, so the sum n4+4n has last digit as 5.
The only part i cannot figure out so far is for n's ending with 5. But a few calculated values seem to show that for n's ending with 5, the expression is still composite.
so most prolly the only solution is n=1.
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rmsgrey
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Re: Prime Time
« Reply #2 on: Nov 29th, 2003, 7:12am » |
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n4+4n=(n2+2n)2-n22n+1
For n odd, n22n+1 is square, so n4+4n is the difference of two squares, so composite, provided n2+2n>n*2(n+1)/2+1
which is certainly true for
2(n-1)/2>n
or n>6
As TenaliRaman pointed out, even n give composite results, so we only need to inspect n=1,3 or 5 giving 5, 145 and 1649(=17*97) respectively.
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TenaliRaman
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Re: Prime Time
« Reply #3 on: Nov 29th, 2003, 7:39am » |
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pretty neat solution rmsgrey !!
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