Primes With Complex Factors
Matthew Seaman
m.seaman at infracaninophile.co.uk
Sun May 11 21:59:56 BST 2008
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Lee Brotherston wrote:
> On Sun, May 11, 2008 at 08:12:07PM +0100, Frank Shute wrote:
>> I'd like to see some higher primes than 2 factored with complex
>> numbers. Two's a bit of a special case (only even prime) and small.
>=20
> Yeah - it would be interesting.
Well, if you're going to multiply two complex numbers and end up with
a positive real prime number as a result, then those numbers must have
equal and opposite phase -- ie. they are complex conjugate to within a
scale factor.
In the general case, we want:
(a + ib)(c + id) =3D p [1]
where p is prime.
ac - bd + i(ad + bc) =3D p
The imaginary part of p is zero, so:
ad + bc =3D 0
a/b =3D -c/d=20
Which is the phase angle requirement above.
This implies that we can rewrite [1] as:
(a + ib)(c + id) =3D R(a + ib)(a - ib) =3D p
where R is just a scale factor. This leads to:
R(a^2 + b^2) =3D p
If we now require all the entities a, b, c, d to be integers then R must
be identically 1, or p could not be prime.
So in order for a prime number to have this sort of complex factorization=
it must be equal to the sum of two squares. There are a number of result=
s
from just the first 10 integers above zero:
1 2 3 4 5 6 7 8 9 10
1 2* 5* 10 17 26 37* 50 65 82 101*
2 8 13* 20 29* 40 53* 68 85 104
3 18 25 34 45 58 73* 90 109
4 32 41* 52 65 80 97* 116=20
5 50 61* 74 89* 106 125
6 72 85 100 117 136
7 98 113* 130 149*
8 128 145 164=20
9 162 181*
10 200
Cheers,
Matthew
--=20
Dr Matthew J Seaman MA, D.Phil. 7 Priory Courtyard
Flat 3
PGP: http://www.infracaninophile.co.uk/pgpkey Ramsgate
Kent, CT11 9PW
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