Prove that there exist infinitely many primes of the form 4x+1 and infinitely many primes of the form 4x-1.
(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 |
re: Too simplistic?
