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Sum of two powers (Posted on 2004-08-12) Difficulty: 4 of 5
If n is an integer, show that n4 + 4n is never a prime for n>1.

See The Solution Submitted by Federico Kereki    
Rating: 4.2500 (4 votes)

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Some Thoughts Incomplete answer | Comment 2 of 7 |

Obviously if n is even, then the expression will be even, thus not a prime.  So we'll only need to consider odd n's.

If n ends with a digit 1, 3, 7, or 9, then n4 will end with digit 1.  4n always end with a 4 as 4n = 4x4n-1, and 4n-1 = 16(n-1)/2 which always end with 6.  So adding them together will end with digit 5, which is always divisible by 5.

Now the problem are those n ends with 5.  I haven't figured that out yet.  My observation is that it always end with a "49" with the 3rd and 4th last digit being ordered multiple of 8, starting from 16 (5), 24 (15), 32 (25), 40 (35), etc...  For those curious,


5  -  1649
15 - xxx2449
25 - xxx3249
35 - xxx4049
45 - xxx4849
55 - xxx5649
65 - xxx6449
75 - xxx7249
85 - xxx8049
95 - xxx8849
105 -xxx9649

This can be proven similar to the way above, setting n = 10k+5.


  Posted by Bon on 2004-08-12 14:46:53
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