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The Q Square of Factoring

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This definition has been superseded by my new definition of Q Factors as a subset of N. See Q Factor Tessellation.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

       

2007-2008 Michael M. Ross