20180530, 17:41  #1 
Jun 2003
Oxford, UK
2×953 Posts 
Sorry if this is obvious
I am wondering if the following holds true for all biprimes comprising different odd factors:
Let B be a biprime with two different odd factors p and q Then at least one n exists such that p==x(mod n) and q==y(mod n) and B == (x*y)(mod n), holds true where:
My reason for thinking this might be worth postulating is based on some work I have done looking at biprimes with odd factors that are similar in size. I carried out an exercise looking at the following 21580 biprimes:
I investigated the range of even n from 16 to 4000 and found at least one n providing a solution for each of the 21580 biprimes  in fact, the average of n found per biprime was about 22. There is an interesting spectrum of results for each n  they are not equal. In fact the graph of solutions for n over a very wide range shows there are ranges of n in which no solutions exist (I think this means they can't exist), and islands of n where solutions do exist. I have also put further restrictions on x and y, for example by insisting x and y are odd primes, (i.e. x or y not equal to 1) and there are biprimes that do not have a solution over the range of n. This is not to say that such solutions do not exist over a wider range of n, I just haven't found them. 
20180530, 18:55  #2 
"Forget I exist"
Jul 2009
Dumbassville
10000010110001_{2} Posts 
If Goldbach's conjecture is true p and q are primes that are the sum of three primes. Makes B a sum of 9 not necessarily distinct Biprimes. Not much else I can say.

20180531, 08:25  #3  
Jun 2003
Oxford, UK
2·953 Posts 
Quote:
Running the same range of n (164000) showed that, in fact, 60 of the biprimes out of 21580 failed to provide a result where x and/or y were 1 or prime. I widened out the range of n, of the 60 fails in the 164000 range, 51 of these provided a result for n=4,6,8,10,12 or 14, so 9 failed to still provide a result (n=2 is trivial.) Widening the range at the other end to include n up to 20000, (i.e. range 420000), then 5 biprimes failed to provide a result. Running right up to n= floor(0.25*H^2*1.0001e9^0.5); where H is p/q  this is the point at which I am not sure further results are possible  provides an annoying count of 1 biprime failing to provide a result. I am not sure that my programming is that great, so maybe it is a possible that the conjecture is still standing. However I will have to restate the conjecture to deal with n=2 Code:
Let B be a biprime with two different odd factors p and q Then at least one n exists such that p==x(mod n) and q==y(mod n) and B == (x*y)(mod n), holds true where:
Last fiddled with by robert44444uk on 20180531 at 08:37 

20180531, 08:49  #4 
Jun 2003
Oxford, UK
2×953 Posts 
Here are two graphs of the numbers of biprimes with results at each n  there are two graphs because the y scales are very skewed.

20180531, 10:10  #5  
Jun 2003
Oxford, UK
2×953 Posts 
Quote:


20180531, 11:55  #6 
"Forget I exist"
Jul 2009
Dumbassville
8,369 Posts 
So only biprimes above 6 with the minimum of 77 if we want both x,y prime.

20180531, 13:16  #7 
Feb 2017
Nowhere
37×103 Posts 
[QUOTE=robert44444uk;488647]I am wondering if the following holds true for all biprimes comprising different odd factors:
Let B be a biprime with two different odd factors p and q Then at least one n exists such that p==x(mod n) and q==y(mod n) and B == (x*y)(mod n), holds true where:

20180531, 13:25  #8 
Jun 2003
Oxford, UK
2·953 Posts 
Thanks for the observations, science_man_88!
For the conjecture to hold, I am pretty certain we need to allow either or both x and y to be 1 as well as prime. Some examples: The biprime 100000194313 only provides one solution, and that is for n=14 where biprime factor 291887 is 5 (mod 14), biprime factor 342599 is 1 (mod 14) biprime 100000194313 is 5 (mod 14) here y =1 We need to go up to n = almost B^0.5 to ensure complete coverage. The smallest of the 2 solutions for biprime 100008836173 is biprime factor 309599 is 23 (mod 309576), biprime factor 323027 is 13451 (mod 309576) biprime 100008836173 is 309373 (mod 309576) At the other end of the scale: biprime 100008007241 provides 176 solutions, of which 158 comprise x and y both prime. 
20180531, 13:29  #9 
Jun 2003
Oxford, UK
2·953 Posts 
Please see post #3, where I acknowledge n>2 is an additional bound. I agree: n=2 is trivial.
Last fiddled with by robert44444uk on 20180531 at 13:30 
20180601, 10:05  #10 
"Forget I exist"
Jul 2009
Dumbassville
20261_{8} Posts 
P=7;q=223 is a counterexample without using n >7 or n=0 you won't find n.
Last fiddled with by science_man_88 on 20180601 at 10:05 
20180601, 10:08  #11 
Jun 2003
Oxford, UK
772_{16} Posts 
As the p/q ratio range tested gets closer to 1, the graph, produced by plotting results of the solutions per n for the 5661 biprimes, sharpens into islands of possible solutions and islands without solutions.
Last fiddled with by robert44444uk on 20180601 at 10:11 
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