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Old 2018-05-30, 17:41   #1
robert44444uk
 
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Default 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:
  • x and y are either 1 or odd primes;
  • x*y <n;
  • n is <p and <q; and
  • n ==0(mod 2)

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:
  • those with different odd factors p and q between 1e9 and1.0001e9
  • and where p/q >1.001 and p/q<2

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.
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Old 2018-05-30, 18:55   #2
science_man_88
 
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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.
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Old 2018-05-31, 08:25   #3
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Quote:
Originally Posted by robert44444uk View Post
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
Hmm, I should have checked my output results first! - my results included non-prime x and y. Also my range of biprimes was 1e11 to 1.0001e11 and not 1e9 to 1.0001e9 !

Running the same range of n (16-4000) 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 16-4000 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 4-20000), 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 re-state 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: 
  • x and y are either 1 or odd primes;
  • x*y <n;
  • n is <p and <q; and
  • n ==0(mod 2)
  • n>2

Last fiddled with by robert44444uk on 2018-05-31 at 08:37
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Old 2018-05-31, 08:49   #4
robert44444uk
 
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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.
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Old 2018-05-31, 10:10   #5
robert44444uk
 
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Quote:
Originally Posted by robert44444uk View Post
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.
I ran the code up to n= floor(1.0001e9^0.5) and all biprimes had at least one result, so the conjecture stills stands.
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Old 2018-05-31, 11:55   #6
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Quote:
Originally Posted by robert44444uk View Post

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: 
  • x and y are either 1 or odd primes;
  • x*y <n;
  • n is <p and <q; and
  • n ==0(mod 2)
  • n>2
So only biprimes above 6 with the minimum of 77 if we want both x,y prime.
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Old 2018-05-31, 13:16   #7
Dr Sardonicus
 
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[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:
  • x and y are either 1 or odd primes;
  • x*y <n;
  • n is <p and <q; and
  • n ==0(mod 2)
n = 2, x = y = 1 appear to fulfill all your requirements.
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Old 2018-05-31, 13:25   #8
robert44444uk
 
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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.
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Old 2018-05-31, 13:29   #9
robert44444uk
 
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Quote:
Originally Posted by Dr Sardonicus View Post
n = 2, x = y = 1 appear to fulfill all your requirements.
Please see post #3, where I acknowledge n>2 is an additional bound. I agree: n=2 is trivial.

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Old 2018-06-01, 10:05   #10
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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 2018-06-01 at 10:05
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Old 2018-06-01, 10:08   #11
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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.
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