The Q Square of Factoring

NATURAL NUMBERS

exploring the undesigned
intelligence of the numberverse

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First published May 24
Revised June 2, 2008

This definition has been superseded by my new definition of Q Factors as a subset of N. See Q Factor Tessellation.

What is a semiprime?
A natural number that is the product of two prime numbers. *

See also Q Factor Tessellation

For every semiprime N there exists a perfect square Q* that is proximate^† to (½*N)² and for which the greatest common divisor of N and Q is a prime factor of N.

* The value of (½*N)² is the 2nd adjacent perfect square of N.

† Q is a value that exceeds (½*N)² by a number of quadratic intervals that is less than ½*sqrt(N).

We can extend this statement to all composites with the caveat that the Q square's GCD will not necessarily be prime.

Sample data includes non-semiprimes (P=prime, S=semiprime, C=all other composites).

	N	F1	F2	Adj2 PS	Q Squares	Proximity* sqrt(Q) / N
P	199961	199961	1	9996200361
S	199963	557	359	9996400324	10032225921	0.500897666
C	199965	13331	15	9996600289	9996800256	0.500007501
P	199967	199967	1	9996800256
C	199969	539	371	9997000225	9997600144	0.500017503
C	199971	1307	153	9997200196	9997400169	0.500007501
S	199973	643	311	9997400169	10028420164	0.500777605
C	199975	475	421	9997600144	9998000100	0.500012502
C	199977	573	349	9997800121	9998000100	0.500007501
S	199979	15383	13	9998000100	9999200016	0.500032503
S	199981	6451	31	9998200081	10001200036	0.500077507
C	199983	623	321	9998400064	9998600049	0.500007501
C	199985	851	235	9998600049	9999000025	0.500012501
S	199987	881	227	9998800036	10021411449	0.500567537
C	199989	823	243	9999000025	9999200016	0.500007500
S	199991	18181	11	9999200016	10000200001	0.500027501
S	199993	4651	43	9999400009	10003600324	0.500107504
C	199995	597	335	9999600004	9999800001	0.500007500
S	199997	28571	7	9999800001	10000400004	0.500017500
P	199999	199999	1	10000000000
C	200001	489	409	10000200001	10000400004	0.500007500
P	200003	200003	1	10000400004
C	200005	905	221	10000600009	10001000025	0.500012500
C	200007	639	313	10000800016	10001000025	0.500007500
P	200009	200009	1	10001000025
S	200011	28573	7	10001200036	10001800081	0.500017499
C	200013	551	363	10001400049	10001600064	0.500007500
C	200015	545	367	10001600064	10002000100	0.500012499
P	200017	200017	1	10001800081
C	200019	1093	183	10002000100	10002200121	0.500007499
S	200021	1439	139	10002200121	10016006400	0.500347464
P	200023	200023	1	10002400144
C	200025	525	381	10002600169	10002800196	0.500007499
S	200027	631	317	10002800196	10034429584	0.500792393
P	200029	200029	1	10003000225
C	200031	669	299	10003200256	10003400289	0.500007499
P	200033	200033	1	10003400289
C	200035	3637	55	10003600324	10004000400	0.500012498
C	200037	509	393	10003800361	10004000400	0.500007499
C	200039	697	287	10004000400	10004600529	0.500017497
P	200041	200041	1	10004200441
C	200043	717	279	10004400484	10004600529	0.500007498
S	200045	40009	5	10004600529	10005000625	0.500012497

*Where 0.5 = (½*N)²

The table reveals that some composites are twinned according to Q squares. This means that there are pairs of consecutive odd composites, sometimes separated by primes, whose factors can be derived using the same Q square - even though the factors themselves are different. For example, a single Q shares a GCD1 with C and a GCD2 with C+2 (or C+4), and so on.

Download (702KB) a comma-delimited analysis of all odd Ns from 135691 to 217357

While you soak in what this might mean, here is some recommended reading...

Primes Is In P

Semiprimes

RSA

The Algorithm: Idiom of Modern Science

The P-versus-NP page