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Primary Problem 2 (Posted on 2012-12-09) Difficulty: 4 of 5
Prove that there exist infinitely many primes of the form 4x+1 and infinitely many primes of the form 4x-1.

No Solution Yet Submitted by Math Man    
Rating: 5.0000 (1 votes)

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re: Too simplistic? | Comment 2 of 4 |
(In reply to Too simplistic? by brianjn)

While all primes larger than 3 are in that form, not all numbers of that form are primes.

Here's a list that follows your plan:

7       5      ..
13      11     ..
19      17     ..
25      23     *.
31      29     ..
37      35     .*
43      41     ..
49      47     *.
55      53     *.
61      59     ..
67      65     .*
73      71     ..
79      77     .*
85      83     *.
91      89     *.
97      95     .*
103     101    ..
109     107    ..
115     113    *.
121     119    **
127     125    .*
133     131    *.
139     137    ..
145     143    **
151     149    ..
157     155    .*
163     161    .*
169     167    *.
175     173    *.
181     179    ..
187     185    **
193     191    ..
199     197    ..
205     203    **
211     209    .*
217     215    **
223     221    .*
229     227    ..
235     233    *.
241     239    ..
247     245    **
253     251    *.
259     257    *.
265     263    *.
271     269    ..
277     275    .*
283     281    ..

In the pairs, if both numbers are prime, they are followed by "..". If either the first or second is composite, the first or second "." is replaced with an asterisk, and if both are composite, then "**".

The double-dots get rarer as you go up. The question is whether the primes (known to be infinitely many) eventually settle only into either .* or *. places, and the double dots eventually stop altogether.

So the idea is to prove that pairs of primes exist in infinite number rather than just isolated primes.


  Posted by Charlie on 2012-12-10 10:15:40
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