John Grieve: Sophie Germain    
 Sophie Germain7 comments
5 Aug 2007 @ 08:22, by John Grieve


Dedicated to Sophie Germain

Sophie Germain primes and primes terminating in recurring 9’s

Sophie Germain primes are primes p where 2p + 1 is also a prime. It is noticeable that it is possible to have a sequence of such primes. The primes of such a form and the sequence that seem most interesting to me are the ones where the Sophie Germain prime terminates in a 9.

It is of great interest that we can get sequences of Sophie Germain primes that end in this way:

89 , 179 , 359 , 719 , 1439 , 2879 (D.Wells 1986 pg 115)

What I would like to assert is that numbers ending in 9s and recurring 9s such as 599 , 1199 , 2399 etc., while not all themselves prime, exhibit an underlying pattern from which the exceptions (those that are not prime) can be explained , and which in fact furnishes one of the few ascertainable forms for prime numbers.

Now, when we double a number and add 1 we are in fact, usually, changing its remainder modulus its divisors (if it has any). But not always. It so happens that the numbers 3 , 5 and 7 when you start with a certain remainder and double it and add 1, you go into a loop which continues indefinitely. Now with the number 5, the remainder that has this remarkable property is 4. Now all odd numbers with 4mod5 in fact terminate in a 9. This explains why we can get 6 Sophie Germain primes in sequence without any of them being divisible by 5.

If we link this remarkable property of the number 5, with similar properties of other divisors, such as 3 and 7, which both exhibit loops on using certain remainders, then we get sequences of numbers which will never be divisible by 3 , 5 or 7. These three divisors account for about two-thirds of composites. Therefore numbers such as 599, 2399 and those ending in recurring 9s of any length, are quite likely to be prime and where they are not it can be explained in the way indicated. There is an obvious link with Cunningham chains , not all of which , however, terminate in 9s.

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7 comments

7 Aug 2007 @ 17:46 by Bob @86.142.238.203 : not good at maths
Sorry, I got confused, will ring you now.  

14 May 2008 @ 21:04 by bbbbbbbbbb @207.224.177.252 : hr
dhhhhhhhhhhhndswmmnuumsssssn  

14 May 2008 @ 21:05 by bob @207.224.177.252 : ew
zbc  

14 May 2008 @ 21:10 by jay @207.224.177.252 : wow
math is so difficult  

14 May 2008 @ 21:12 by won jin @207.224.177.252 : sohie
i absulutely love sophies work! BIG FAN  

14 May 2008 @ 21:13 by mc dre @207.224.177.252 : yo
math is hard  

14 May 2008 @ 21:14 by susana @207.224.177.252 : aha
math is funn  

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