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        riddles >> easy >> Prime Time
        
(Message started by: THUDandBLUNDER on Nov 29^th, 2003, 4:17am)

Title: Prime Time
Post by THUDandBLUNDER on Nov 29^th, 2003, 4:17am

For which positive integer values of n is n⁴ + 4ⁿ prime?

Title: Re: Prime Time
Post by TenaliRaman on Nov 29^th, 2003, 6:38am

(not a complete solution)
::[hide]
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.
[/hide]::

Title: Re: Prime Time
Post by rmsgrey on Nov 29^th, 2003, 7:12am

::[hide]
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.
[/hide]::

Title: Re: Prime Time
Post by TenaliRaman on Nov 29^th, 2003, 7:39am

pretty neat solution rmsgrey !!