## Prime Game

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• 7,31,109,349,1061,103,29,97,43,13 11,41,131,37 17,59 If you sum 3 consecutive odd prime numbers p,q,r, you get a number s which is either prime or not:
Message 1 of 4 , Jan 2, 2006
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7,31,109,349,1061,103,29,97,43,13
11,41,131,37
17,59

If you sum 3 consecutive odd prime numbers p,q,r, you get a number s
which is either prime or not: p+q+r=s. If s is prime, you call it p
and repeat the game. If s is not prime, you call the greatest prime
factor p and repeat the game. Finally, you get into an infinite
cycle, which is one of the above 3 series, no matter what prime
numbers you choose. Example: p=7

7+11+13=31
31+37+41=109
109+113+127=349
349+353+359=1061
1061+1063+1069=3193=31*103
103+107+109=319=11*29
29+31+37=97
97+101+103=301=7*43
43+47+53=143=11*13
13+17+19=49=7*7
7+11+13

Questions:

1.) Why doesn't the series diverge? Proof?
2.) Why are there these 3 series and not 4 or 5? Proof?
3.) Instead of s=p+q+r try s=p²+q²+r².
• From: Werner D. Sand Date: 01/02/06 09:30:35 To: primenumbers@yahoogroups.com Subject: [PrimeNumbers] Prime Game 7,31,109,349,1061,103,29,97,43,13 11,41,131,37
Message 2 of 4 , Jan 2, 2006
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From: Werner D. Sand
Date: 01/02/06 09:30:35

7,31,109,349,1061,103,29,97,43,13
11,41,131,37
17,59

If you sum 3 consecutive odd prime numbers p,q,r, you get a number s
which is either prime or not: p+q+r=s. If s is prime, you call it p
and repeat the game. If s is not prime, you call the greatest prime
factor p and repeat the game. Finally, you get into an infinite
cycle, which is one of the above 3 series, no matter what prime
numbers you choose. Example: p=7

7+11+13=31
31+37+41=109
109+113+127=349
349+353+359=1061
1061+1063+1069=3193=31*103
103+107+109=319=11*29
29+31+37=97
97+101+103=301=7*43
43+47+53=143=11*13
13+17+19=49=7*7
7+11+13

Questions:

1.) Why doesn't the series diverge? Proof?

Kermit says:

I started with 113. How far would I have to go to end up with one of the
3 series?

I suspect that the termination into one of those three series has something
to
do with every prime >3 being of the form 6N+1 or 6N-1.

[Non-text portions of this message have been removed]
• Hi Kermit, after 3 steps you are in the 2nd cycle: 113+127+131=371=7*53 53+59+61=173 173+179+181=533=13*41 41... Werner
Message 3 of 4 , Jan 3, 2006
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Hi Kermit,

after 3 steps you are in the 2nd cycle:

113+127+131=371=7*53
53+59+61=173
173+179+181=533=13*41
41...

Werner
• Guys, I call it a game , but it s mathematics!
Message 4 of 4 , Jan 4, 2006
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Guys, I call it a "game", but it's mathematics!
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