| T.R | Title | User | Personal
Name | Date | Lines |
|---|
| 1804.1 | | HANNAH::OSMAN | see HANNAH::IGLOO$:[OSMAN]ERIC.VT240 | Wed Oct 06 1993 13:54 | 16 |
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> 4 n
>Prove that n + 4 is non-prime for all n > 1.
n^4 + 4^n = n^4 + (2^n)^2 = (n^2)^2 + (2^n)^2 = a^2 + b^2
Am I close ?
/Eric
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| 1804.2 | | RUSURE::EDP | Always mount a scratch monkey. | Wed Oct 06 1993 16:34 | 29 |
| Re .0:
Solution found with help of computer algebra:
Divide n by 10, yielding n = 10*q + r.
If r is even, then n^4+4^n is clearly even and greater than 2, so it is
composite.
If r is 1, 3, 7, or 9, then n^4 is congruent to 1 modulo 5. And n is
odd for these r, so 4^n is congruent to 4 modulo 5, so n^4+4^n is
divisible by 5 and clearly greater than 5.
If r is 5, then n^4+4^n = (10*q+5)^4 + 4^(10*q+5) =
[ 2^(10*q+5) - 5*(2q+1)*2^(5q+3) + 25(2q+1)^2 ] *
[ 2^(10*q+5) + 5*(2q+1)*2^(5q+3) + 25(2q+1)^2 ].
Both factors are clearly greater than one when q is non-negative, so
n^4+4^n is composite in all cases when n>1.
Derive provided the factorization easily; Maple did not.
-- edp
Public key fingerprint: 8e ad 63 61 ba 0c 26 86 32 0a 7d 28 db e7 6f 75
To get PGP, FTP /pub/unix/security/crypt/pgp23A.zip from nic.funet.fi.
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| 1804.3 | | AUSSIE::GARSON | Hotel Garson: No Vacancies | Wed Oct 06 1993 21:48 | 71 |
| 1804.4 | | RUSURE::EDP | Always mount a scratch monkey. | Thu Oct 07 1993 11:42 | 11 |
| Re .3:
> I don't know whether you can persuade Derive to spot it.
Yes, when I enter n^4+4^n and substitute 2j+1 for n, Derive factors the
result easily. It works surprisingly well in factoring and simplifying
expressions, handling cases that Maple does not or that Maple is
awkward at.
-- edp
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| 1804.5 | .3 is obvious in hindsight! | TROOA::RITCHE | From the desk of Allen Ritche... | Thu Oct 07 1993 17:47 | 22 |
| 1804.6 | | RUSURE::EDP | Always mount a scratch monkey. | Thu Oct 07 1993 19:03 | 17 |
| Re .5:
Derive is a symbolic math program available from Soft Warehouse, 3660
Waialae Avenue, Suite 304, Honolulu, HI, 96816-3236. It is also sold
by Educalc, 1-800-677-7001. It runs under DOS on PCs, including the
HP-95 palmtop. It has an excellent character-based interface; it
manages to draw expressions reasonably well. It's not as fancy as
Maple with a Windows interface, and it doesn't have the humongous
library of functions, but it actually does better than Maple and
Mathematica at many factorizations and simplifications.
-- edp
Public key fingerprint: 8e ad 63 61 ba 0c 26 86 32 0a 7d 28 db e7 6f 75
To get PGP, FTP /pub/unix/security/crypt/pgp23A.zip from nic.funet.fi.
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| 1804.7 | | AUSSIE::GARSON | Hotel Garson: No Vacancies | Fri Oct 08 1993 02:01 | 31 |
| 1804.8 | yup | HERON::BUCHANAN | The was not found. | Fri Oct 08 1993 10:01 | 27 |
| 1804.9 | | CFSCTC::GILBERT | | Fri Oct 08 1993 16:59 | 3 |
| What about n^6 + 6^n ?
What about n^5 + 5^n ? Does it generate an infinite number of primes?
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| 1804.10 | | AUSSIE::GARSON | Hotel Garson: No Vacancies | Sat Oct 09 1993 03:10 | 50
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