https://www.explainxkcd.com/wiki/index.php?title=1310:_Goldbach_Conjectures&feed=atom&action=history1310: Goldbach Conjectures - Revision history2024-03-28T08:47:33ZRevision history for this page on the wikiMediaWiki 1.30.0https://www.explainxkcd.com/wiki/index.php?title=1310:_Goldbach_Conjectures&diff=334248&oldid=prev172.70.91.61: Undo revision 334242 by B for brain (talk) I think this point needs rewriting, for several reasons. I can see where it started going, but not fully thought through how far it goes. + &->and!2024-02-05T17:16:48Z<p>Undo revision 334242 by <a href="/wiki/index.php/Special:Contributions/B_for_brain" title="Special:Contributions/B for brain">B for brain</a> (<a href="/wiki/index.php/User_talk:B_for_brain" title="User talk:B for brain">talk</a>) I think this point needs rewriting, for several reasons. I can see where it started going, but not fully thought through how far it goes. + &->and!</p>
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<tr><td class='diff-marker'> </td><td style="background-color: #f9f9f9; color: #333333; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #e6e6e6; vertical-align: top; white-space: pre-wrap;"><div>The title text gives the same treatment to the {{w|Twin prime|twin prime conjecture}}, which says that there are infinitely many pairs of primes ''where one is 2 more than the other'' (e.g. 3 and 5). The title text adds a "weak" conjecture, according to which there are simply infinitely many pairs of primes (with no mention of the distance between them). This is true; {{w|Euclid's theorem}} says that there are an infinite number of primes, and so you can simply pick any two (e.g. 5 and 13) and call them a pair.</div></td><td class='diff-marker'> </td><td style="background-color: #f9f9f9; color: #333333; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #e6e6e6; vertical-align: top; white-space: pre-wrap;"><div>The title text gives the same treatment to the {{w|Twin prime|twin prime conjecture}}, which says that there are infinitely many pairs of primes ''where one is 2 more than the other'' (e.g. 3 and 5). The title text adds a "weak" conjecture, according to which there are simply infinitely many pairs of primes (with no mention of the distance between them). This is true; {{w|Euclid's theorem}} says that there are an infinite number of primes, and so you can simply pick any two (e.g. 5 and 13) and call them a pair.</div></td></tr>
<tr><td class='diff-marker'> </td><td style="background-color: #f9f9f9; color: #333333; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #e6e6e6; vertical-align: top; white-space: pre-wrap;"></td><td class='diff-marker'> </td><td style="background-color: #f9f9f9; color: #333333; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #e6e6e6; vertical-align: top; white-space: pre-wrap;"></td></tr>
<tr><td class='diff-marker'>−</td><td style="color:black; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div>It also adds a "strong" conjecture where ''every'' prime is now a twin prime. This is easily proven false; 23 is prime, for example, but cannot be one of a pair as neither 21 nor 25 are. However, Randall adds a humorous {{w|hedge (linguistics)|hedge}} that some prime numbers "may not look prime at first". <del class="diffchange diffchange-inline">(However, because of how the title text is worded, it technicaly means all numbers, except for 1, 0, and the negative & complex numbers, are prime.)</del></div></td><td class='diff-marker'>+</td><td style="color:black; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div>It also adds a "strong" conjecture where ''every'' prime is now a twin prime. This is easily proven false; 23 is prime, for example, but cannot be one of a pair as neither 21 nor 25 are. However, Randall adds a humorous {{w|hedge (linguistics)|hedge}} that some prime numbers "may not look prime at first".</div></td></tr>
<tr><td class='diff-marker'> </td><td style="background-color: #f9f9f9; color: #333333; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #e6e6e6; vertical-align: top; white-space: pre-wrap;"></td><td class='diff-marker'> </td><td style="background-color: #f9f9f9; color: #333333; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #e6e6e6; vertical-align: top; white-space: pre-wrap;"></td></tr>
<tr><td class='diff-marker'> </td><td style="background-color: #f9f9f9; color: #333333; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #e6e6e6; vertical-align: top; white-space: pre-wrap;"><div>Lastly, the tautological prime conjecture states that it itself is true while making no statement about primes. It is not technically a {{w|tautology}} but more of a plain assertion. Randall has mentioned tautologies before in [[703: Honor Societies]].</div></td><td class='diff-marker'> </td><td style="background-color: #f9f9f9; color: #333333; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #e6e6e6; vertical-align: top; white-space: pre-wrap;"><div>Lastly, the tautological prime conjecture states that it itself is true while making no statement about primes. It is not technically a {{w|tautology}} but more of a plain assertion. Randall has mentioned tautologies before in [[703: Honor Societies]].</div></td></tr>
</table>172.70.91.61https://www.explainxkcd.com/wiki/index.php?title=1310:_Goldbach_Conjectures&diff=334242&oldid=prevB for brain: /* Explanation */2024-02-05T15:00:27Z<p><span dir="auto"><span class="autocomment">Explanation</span></span></p>
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<tr><td class='diff-marker'> </td><td style="background-color: #f9f9f9; color: #333333; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #e6e6e6; vertical-align: top; white-space: pre-wrap;"><div>The title text gives the same treatment to the {{w|Twin prime|twin prime conjecture}}, which says that there are infinitely many pairs of primes ''where one is 2 more than the other'' (e.g. 3 and 5). The title text adds a "weak" conjecture, according to which there are simply infinitely many pairs of primes (with no mention of the distance between them). This is true; {{w|Euclid's theorem}} says that there are an infinite number of primes, and so you can simply pick any two (e.g. 5 and 13) and call them a pair.</div></td><td class='diff-marker'> </td><td style="background-color: #f9f9f9; color: #333333; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #e6e6e6; vertical-align: top; white-space: pre-wrap;"><div>The title text gives the same treatment to the {{w|Twin prime|twin prime conjecture}}, which says that there are infinitely many pairs of primes ''where one is 2 more than the other'' (e.g. 3 and 5). The title text adds a "weak" conjecture, according to which there are simply infinitely many pairs of primes (with no mention of the distance between them). This is true; {{w|Euclid's theorem}} says that there are an infinite number of primes, and so you can simply pick any two (e.g. 5 and 13) and call them a pair.</div></td></tr>
<tr><td class='diff-marker'> </td><td style="background-color: #f9f9f9; color: #333333; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #e6e6e6; vertical-align: top; white-space: pre-wrap;"></td><td class='diff-marker'> </td><td style="background-color: #f9f9f9; color: #333333; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #e6e6e6; vertical-align: top; white-space: pre-wrap;"></td></tr>
<tr><td class='diff-marker'>−</td><td style="color:black; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div>It also adds a "strong" conjecture where ''every'' prime is now a twin prime. This is easily proven false; 23 is prime, for example, but cannot be one of a pair as neither 21 nor 25 are. However, Randall adds a humorous {{w|hedge (linguistics)|hedge}} that some prime numbers "may not look prime at first".</div></td><td class='diff-marker'>+</td><td style="color:black; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div>It also adds a "strong" conjecture where ''every'' prime is now a twin prime. This is easily proven false; 23 is prime, for example, but cannot be one of a pair as neither 21 nor 25 are. However, Randall adds a humorous {{w|hedge (linguistics)|hedge}} that some prime numbers "may not look prime at first". <ins class="diffchange diffchange-inline">(However, because of how the title text is worded, it technicaly means all numbers, except for 1, 0, and the negative & complex numbers, are prime.)</ins></div></td></tr>
<tr><td class='diff-marker'> </td><td style="background-color: #f9f9f9; color: #333333; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #e6e6e6; vertical-align: top; white-space: pre-wrap;"></td><td class='diff-marker'> </td><td style="background-color: #f9f9f9; color: #333333; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #e6e6e6; vertical-align: top; white-space: pre-wrap;"></td></tr>
<tr><td class='diff-marker'> </td><td style="background-color: #f9f9f9; color: #333333; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #e6e6e6; vertical-align: top; white-space: pre-wrap;"><div>Lastly, the tautological prime conjecture states that it itself is true while making no statement about primes. It is not technically a {{w|tautology}} but more of a plain assertion. Randall has mentioned tautologies before in [[703: Honor Societies]].</div></td><td class='diff-marker'> </td><td style="background-color: #f9f9f9; color: #333333; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #e6e6e6; vertical-align: top; white-space: pre-wrap;"><div>Lastly, the tautological prime conjecture states that it itself is true while making no statement about primes. It is not technically a {{w|tautology}} but more of a plain assertion. Randall has mentioned tautologies before in [[703: Honor Societies]].</div></td></tr>
</table>B for brainhttps://www.explainxkcd.com/wiki/index.php?title=1310:_Goldbach_Conjectures&diff=324230&oldid=prev172.70.92.163: Restored joke citation needed template2023-09-23T09:45:54Z<p>Restored joke citation needed template</p>
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</tr><tr><td colspan="2" class="diff-lineno" id="mw-diff-left-l17" >Line 17:</td>
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<tr><td class='diff-marker'> </td><td style="background-color: #f9f9f9; color: #333333; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #e6e6e6; vertical-align: top; white-space: pre-wrap;"><div>* The "very strong" conjecture says that every odd number is prime. This is false, because some odd numbers are {{w|Composite_number|composite}} (e.g. 9, 15, 21), and composite numbers are not prime.{{citation needed}} But if this conjecture ''were'' true, it would make Goldbach's (strong) conjecture true as well, because every even number can be written as the sum of two odd numbers (which, by this "conjecture", are prime).</div></td><td class='diff-marker'> </td><td style="background-color: #f9f9f9; color: #333333; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #e6e6e6; vertical-align: top; white-space: pre-wrap;"><div>* The "very strong" conjecture says that every odd number is prime. This is false, because some odd numbers are {{w|Composite_number|composite}} (e.g. 9, 15, 21), and composite numbers are not prime.{{citation needed}} But if this conjecture ''were'' true, it would make Goldbach's (strong) conjecture true as well, because every even number can be written as the sum of two odd numbers (which, by this "conjecture", are prime).</div></td></tr>
<tr><td class='diff-marker'> </td><td style="background-color: #f9f9f9; color: #333333; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #e6e6e6; vertical-align: top; white-space: pre-wrap;"><div>* The "extremely strong" conjecture says that numbers stop at 7. {{w|8|This is false}}, but if it ''were'' true, it might make the above conjecture true as well: 9 is the first odd composite number, so stopping at 7 would eliminate all odd composite numbers. (1 is neither prime nor composite, but it ''has'' been counted as a prime number in the past. Randall may have meant 1 to be an unspoken exception, or he may be returning to the older definition that included 1 as prime.)</div></td><td class='diff-marker'> </td><td style="background-color: #f9f9f9; color: #333333; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #e6e6e6; vertical-align: top; white-space: pre-wrap;"><div>* The "extremely strong" conjecture says that numbers stop at 7. {{w|8|This is false}}, but if it ''were'' true, it might make the above conjecture true as well: 9 is the first odd composite number, so stopping at 7 would eliminate all odd composite numbers. (1 is neither prime nor composite, but it ''has'' been counted as a prime number in the past. Randall may have meant 1 to be an unspoken exception, or he may be returning to the older definition that included 1 as prime.)</div></td></tr>
<tr><td class='diff-marker'>−</td><td style="color:black; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div>* In the other direction, the "very weak" conjecture says that every number above 7 can be written as the sum of two other numbers. This is true, but as it says nothing about primes, it isn't enough to prove Goldbach's weak conjecture. The weak conjecture being true would automatically make this one true, though (if we didn't already know it was true).</div></td><td class='diff-marker'>+</td><td style="color:black; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div>* In the other direction, the "very weak" conjecture says that every number above 7 can be written as the sum of two other numbers. This is true,<ins class="diffchange diffchange-inline">{{citation needed}} </ins>but as it says nothing about primes, it isn't enough to prove Goldbach's weak conjecture. The weak conjecture being true would automatically make this one true, though (if we didn't already know it was true).</div></td></tr>
<tr><td class='diff-marker'> </td><td style="background-color: #f9f9f9; color: #333333; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #e6e6e6; vertical-align: top; white-space: pre-wrap;"><div>* The "extremely weak" conjecture says that "numbers just keep going". This is true, but it may not actually be implied by the above conjectures. Those say that numbers above 7 have certain properties, without ''requiring'' that such numbers exist. This may seem like a nitpicky point, but mathematicians love those; it also causes problems, because the "extremely strong" and "extremely weak" conjectures contradict each other. If the other conjectures were rewritten to say "these numbers exist, ''and'' have these properties", then they would imply this "extremely weak" conjecture, but then the "extremely strong" one would have to be stricken off.</div></td><td class='diff-marker'> </td><td style="background-color: #f9f9f9; color: #333333; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #e6e6e6; vertical-align: top; white-space: pre-wrap;"><div>* The "extremely weak" conjecture says that "numbers just keep going". This is true, but it may not actually be implied by the above conjectures. Those say that numbers above 7 have certain properties, without ''requiring'' that such numbers exist. This may seem like a nitpicky point, but mathematicians love those; it also causes problems, because the "extremely strong" and "extremely weak" conjectures contradict each other. If the other conjectures were rewritten to say "these numbers exist, ''and'' have these properties", then they would imply this "extremely weak" conjecture, but then the "extremely strong" one would have to be stricken off.</div></td></tr>
<tr><td class='diff-marker'> </td><td style="background-color: #f9f9f9; color: #333333; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #e6e6e6; vertical-align: top; white-space: pre-wrap;"></td><td class='diff-marker'> </td><td style="background-color: #f9f9f9; color: #333333; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #e6e6e6; vertical-align: top; white-space: pre-wrap;"></td></tr>
</table>172.70.92.163https://www.explainxkcd.com/wiki/index.php?title=1310:_Goldbach_Conjectures&diff=324229&oldid=prev162.158.189.112: /* Explanation */ I think It is trivially obvious that every number greater than 7 is able to be written as the sum of two other numbers.2023-09-23T09:18:58Z<p><span dir="auto"><span class="autocomment">Explanation: </span> I think It is trivially obvious that every number greater than 7 is able to be written as the sum of two other numbers.</span></p>
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</tr><tr><td colspan="2" class="diff-lineno" id="mw-diff-left-l17" >Line 17:</td>
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<tr><td class='diff-marker'> </td><td style="background-color: #f9f9f9; color: #333333; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #e6e6e6; vertical-align: top; white-space: pre-wrap;"><div>* The "very strong" conjecture says that every odd number is prime. This is false, because some odd numbers are {{w|Composite_number|composite}} (e.g. 9, 15, 21), and composite numbers are not prime.{{citation needed}} But if this conjecture ''were'' true, it would make Goldbach's (strong) conjecture true as well, because every even number can be written as the sum of two odd numbers (which, by this "conjecture", are prime).</div></td><td class='diff-marker'> </td><td style="background-color: #f9f9f9; color: #333333; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #e6e6e6; vertical-align: top; white-space: pre-wrap;"><div>* The "very strong" conjecture says that every odd number is prime. This is false, because some odd numbers are {{w|Composite_number|composite}} (e.g. 9, 15, 21), and composite numbers are not prime.{{citation needed}} But if this conjecture ''were'' true, it would make Goldbach's (strong) conjecture true as well, because every even number can be written as the sum of two odd numbers (which, by this "conjecture", are prime).</div></td></tr>
<tr><td class='diff-marker'> </td><td style="background-color: #f9f9f9; color: #333333; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #e6e6e6; vertical-align: top; white-space: pre-wrap;"><div>* The "extremely strong" conjecture says that numbers stop at 7. {{w|8|This is false}}, but if it ''were'' true, it might make the above conjecture true as well: 9 is the first odd composite number, so stopping at 7 would eliminate all odd composite numbers. (1 is neither prime nor composite, but it ''has'' been counted as a prime number in the past. Randall may have meant 1 to be an unspoken exception, or he may be returning to the older definition that included 1 as prime.)</div></td><td class='diff-marker'> </td><td style="background-color: #f9f9f9; color: #333333; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #e6e6e6; vertical-align: top; white-space: pre-wrap;"><div>* The "extremely strong" conjecture says that numbers stop at 7. {{w|8|This is false}}, but if it ''were'' true, it might make the above conjecture true as well: 9 is the first odd composite number, so stopping at 7 would eliminate all odd composite numbers. (1 is neither prime nor composite, but it ''has'' been counted as a prime number in the past. Randall may have meant 1 to be an unspoken exception, or he may be returning to the older definition that included 1 as prime.)</div></td></tr>
<tr><td class='diff-marker'>−</td><td style="color:black; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div>* In the other direction, the "very weak" conjecture says that every number above 7 can be written as the sum of two other numbers. This is true,<del class="diffchange diffchange-inline">{{Citation needed}} </del>but as it says nothing about primes, it isn't enough to prove Goldbach's weak conjecture. The weak conjecture being true would automatically make this one true, though (if we didn't already know it was true).</div></td><td class='diff-marker'>+</td><td style="color:black; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div>* In the other direction, the "very weak" conjecture says that every number above 7 can be written as the sum of two other numbers. This is true, but as it says nothing about primes, it isn't enough to prove Goldbach's weak conjecture. The weak conjecture being true would automatically make this one true, though (if we didn't already know it was true).</div></td></tr>
<tr><td class='diff-marker'> </td><td style="background-color: #f9f9f9; color: #333333; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #e6e6e6; vertical-align: top; white-space: pre-wrap;"><div>* The "extremely weak" conjecture says that "numbers just keep going". This is true, but it may not actually be implied by the above conjectures. Those say that numbers above 7 have certain properties, without ''requiring'' that such numbers exist. This may seem like a nitpicky point, but mathematicians love those; it also causes problems, because the "extremely strong" and "extremely weak" conjectures contradict each other. If the other conjectures were rewritten to say "these numbers exist, ''and'' have these properties", then they would imply this "extremely weak" conjecture, but then the "extremely strong" one would have to be stricken off.</div></td><td class='diff-marker'> </td><td style="background-color: #f9f9f9; color: #333333; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #e6e6e6; vertical-align: top; white-space: pre-wrap;"><div>* The "extremely weak" conjecture says that "numbers just keep going". This is true, but it may not actually be implied by the above conjectures. Those say that numbers above 7 have certain properties, without ''requiring'' that such numbers exist. This may seem like a nitpicky point, but mathematicians love those; it also causes problems, because the "extremely strong" and "extremely weak" conjectures contradict each other. If the other conjectures were rewritten to say "these numbers exist, ''and'' have these properties", then they would imply this "extremely weak" conjecture, but then the "extremely strong" one would have to be stricken off.</div></td></tr>
<tr><td class='diff-marker'> </td><td style="background-color: #f9f9f9; color: #333333; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #e6e6e6; vertical-align: top; white-space: pre-wrap;"></td><td class='diff-marker'> </td><td style="background-color: #f9f9f9; color: #333333; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #e6e6e6; vertical-align: top; white-space: pre-wrap;"></td></tr>
</table>162.158.189.112https://www.explainxkcd.com/wiki/index.php?title=1310:_Goldbach_Conjectures&diff=308924&oldid=prev172.70.115.71: added a {{citation needed}}2023-03-21T00:47:33Z<p>added a {{citation needed}}</p>
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<td colspan="2" style="background-color: white; color:black; text-align: center;">Revision as of 00:47, 21 March 2023</td>
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<tr><td class='diff-marker'> </td><td style="background-color: #f9f9f9; color: #333333; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #e6e6e6; vertical-align: top; white-space: pre-wrap;"></td><td class='diff-marker'> </td><td style="background-color: #f9f9f9; color: #333333; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #e6e6e6; vertical-align: top; white-space: pre-wrap;"></td></tr>
<tr><td class='diff-marker'> </td><td style="background-color: #f9f9f9; color: #333333; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #e6e6e6; vertical-align: top; white-space: pre-wrap;"><div>Randall's further conjectures extend this to a whole series of progressively "weaker" and "stronger" statements. His weak conjectures are so weak that they are obviously true; his strong conjectures are so restrictive that they are obviously false. However, for the most part, they really do maintain a weak-strong relationship.</div></td><td class='diff-marker'> </td><td style="background-color: #f9f9f9; color: #333333; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #e6e6e6; vertical-align: top; white-space: pre-wrap;"><div>Randall's further conjectures extend this to a whole series of progressively "weaker" and "stronger" statements. His weak conjectures are so weak that they are obviously true; his strong conjectures are so restrictive that they are obviously false. However, for the most part, they really do maintain a weak-strong relationship.</div></td></tr>
<tr><td class='diff-marker'>−</td><td style="color:black; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div>* The "very strong" conjecture says that every odd number is prime. This is false, because some odd numbers are {{w|Composite_number|composite}} (e.g. 9, 15, 21), and composite numbers are not prime. But if this conjecture ''were'' true, it would make Goldbach's (strong) conjecture true as well, because every even number can be written as the sum of two odd numbers (which, by this "conjecture", are prime).</div></td><td class='diff-marker'>+</td><td style="color:black; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div>* The "very strong" conjecture says that every odd number is prime. This is false, because some odd numbers are {{w|Composite_number|composite}} (e.g. 9, 15, 21), and composite numbers are not prime.<ins class="diffchange diffchange-inline">{{citation needed}} </ins>But if this conjecture ''were'' true, it would make Goldbach's (strong) conjecture true as well, because every even number can be written as the sum of two odd numbers (which, by this "conjecture", are prime).</div></td></tr>
<tr><td class='diff-marker'> </td><td style="background-color: #f9f9f9; color: #333333; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #e6e6e6; vertical-align: top; white-space: pre-wrap;"><div>* The "extremely strong" conjecture says that numbers stop at 7. {{w|8|This is false}}, but if it ''were'' true, it might make the above conjecture true as well: 9 is the first odd composite number, so stopping at 7 would eliminate all odd composite numbers. (1 is neither prime nor composite, but it ''has'' been counted as a prime number in the past. Randall may have meant 1 to be an unspoken exception, or he may be returning to the older definition that included 1 as prime.)</div></td><td class='diff-marker'> </td><td style="background-color: #f9f9f9; color: #333333; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #e6e6e6; vertical-align: top; white-space: pre-wrap;"><div>* The "extremely strong" conjecture says that numbers stop at 7. {{w|8|This is false}}, but if it ''were'' true, it might make the above conjecture true as well: 9 is the first odd composite number, so stopping at 7 would eliminate all odd composite numbers. (1 is neither prime nor composite, but it ''has'' been counted as a prime number in the past. Randall may have meant 1 to be an unspoken exception, or he may be returning to the older definition that included 1 as prime.)</div></td></tr>
<tr><td class='diff-marker'> </td><td style="background-color: #f9f9f9; color: #333333; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #e6e6e6; vertical-align: top; white-space: pre-wrap;"><div>* In the other direction, the "very weak" conjecture says that every number above 7 can be written as the sum of two other numbers. This is true,{{Citation needed}} but as it says nothing about primes, it isn't enough to prove Goldbach's weak conjecture. The weak conjecture being true would automatically make this one true, though (if we didn't already know it was true).</div></td><td class='diff-marker'> </td><td style="background-color: #f9f9f9; color: #333333; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #e6e6e6; vertical-align: top; white-space: pre-wrap;"><div>* In the other direction, the "very weak" conjecture says that every number above 7 can be written as the sum of two other numbers. This is true,{{Citation needed}} but as it says nothing about primes, it isn't enough to prove Goldbach's weak conjecture. The weak conjecture being true would automatically make this one true, though (if we didn't already know it was true).</div></td></tr>
</table>172.70.115.71https://www.explainxkcd.com/wiki/index.php?title=1310:_Goldbach_Conjectures&diff=249690&oldid=prevJacky720: rv2022-05-04T21:31:23Z<p>rv</p>
<a href="//www.explainxkcd.com/wiki/index.php?title=1310:_Goldbach_Conjectures&diff=249690&oldid=249678">Show changes</a>Jacky720https://www.explainxkcd.com/wiki/index.php?title=1310:_Goldbach_Conjectures&diff=249678&oldid=prevEx Kay Cee Dee at 21:31, 4 May 20222022-05-04T21:31:21Z<p></p>
<a href="//www.explainxkcd.com/wiki/index.php?title=1310:_Goldbach_Conjectures&diff=249678&oldid=249312">Show changes</a>Ex Kay Cee Deehttps://www.explainxkcd.com/wiki/index.php?title=1310:_Goldbach_Conjectures&diff=249312&oldid=prevJacky720: rv2022-05-04T21:28:51Z<p>rv</p>
<a href="//www.explainxkcd.com/wiki/index.php?title=1310:_Goldbach_Conjectures&diff=249312&oldid=249277">Show changes</a>Jacky720https://www.explainxkcd.com/wiki/index.php?title=1310:_Goldbach_Conjectures&diff=249277&oldid=prevEx Kay Cee Dee at 21:28, 4 May 20222022-05-04T21:28:30Z<p></p>
<a href="//www.explainxkcd.com/wiki/index.php?title=1310:_Goldbach_Conjectures&diff=249277&oldid=245276">Show changes</a>Ex Kay Cee Deehttps://www.explainxkcd.com/wiki/index.php?title=1310:_Goldbach_Conjectures&diff=245276&oldid=prevJacky720: rv2022-05-04T20:58:12Z<p>rv</p>
<a href="//www.explainxkcd.com/wiki/index.php?title=1310:_Goldbach_Conjectures&diff=245276&oldid=243931">Show changes</a>Jacky720