https://michaelnielsen.org/polymath/index.php?action=history&feed=atom&title=Side_Proof_1 2026-06-29T14:32:09Z Revision history for this page on the wiki MediaWiki 1.42.3 https://michaelnielsen.org/polymath/index.php?title=Side_Proof_1&diff=9627&oldid=prev 2015-05-21T09:26:26Z

<p>Created page with &quot;This page will handle one of the long cases in the <a href="/polymath/index.php?title=Human_proof_that_completely_multiplicative_sequences_have_discrepancy_greater_than_3" title="Human proof that completely multiplicative sequences have discrepancy greater than 3">Human proof that completely multiplicative sequences have discrepancy greater than 3</a>, so that the page can be shorter an...&quot;</p> <p><b>New page</b></p><div>This page will handle one of the long cases in the [[Human proof that completely multiplicative sequences have discrepancy greater than 3]], so that the page can be shorter and not have so many long sections. Specifically, this page will take care of the case where we assume: f(2)=f(7)=f(19)=f(23)=f(29)=1.<br /> <br /> Looking at the table:<br /> <br /> 0 1 2 3 4 5 6 7 8 9<br /> <br /> 0|+ + - + - - + + + 0-9<br /> - - - - + + + - + + 10-19<br /> - - - + - + - - +|+ 20-29<br /> + ? + + - - + ? + + 30-39<br /> - ? - ? - - + ? - + 40-49<br /> + + - ? - + + - + ? 50-59<br /> + ? ? + + + + ? - - 60-69<br /> <br /> We can first see that the discrepancy up to 32 is 3+f(31), so f(31)=-1. Also, there is a cut after 64, because of f[63,66].<br /> <br /> 0 1 2 3 4 5 6 7 8 9<br /> <br /> 0|+ + - + - - + + + 0-9<br /> - - - - + + + - + + 10-19<br /> - - - + - + - - +|+ 20-29<br /> + - + + - - + ? + + 30-39<br /> - ? - ? - - + ? - + 40-49<br /> + + - ? - + + - + ? 50-59<br /> + ? - + +|+ + ? - - 60-69<br /> <br /> Now the discrepancy up to 38 is 3+f(37), therefore f(37)=-1.<br /> <br /> 0 1 2 3 4 5 6 7 8 9<br /> <br /> 0|+ + - + - - + + + 0-9<br /> - - - - + + + - + + 10-19<br /> - - - + - + - - +|+ 20-29<br /> + - + + - - + - + + 30-39<br /> - ? - ? - - + ? - + 40-49<br /> + + - ? - + + - + ? 50-59<br /> + ? - + +|+ + ? - - 60-69<br /> <br /> Here is where things start to get interesting. The discrepancy up to 64, which is 4+f(41)+f(43)+f(47)+f(53)+f(59)+f(61), must be zero because of the cut. Therefore exactly one of f(41), f(43), f(47), f(53), f(59), f(61) must be 1, and the others -1. However, the discrepancy f[243,248] = -4-f(41)+f(61), so if f(41) = 1, f(61) must be -1, but then the discrepancy f[243,248] = -6, forcing the discrepancy to be greater than 3. Therefore f(41)=-1.<br /> <br /> 0 1 2 3 4 5 6 7 8 9<br /> <br /> 0|+ + - + - - + + + 0-9<br /> - - - - + + + - + + 10-19<br /> - - - + - + - - + + 20-29<br /> + - + + - - + - + + 30-39<br /> - - -|? - - + ? - + 40-49<br /> + + - ? - + + - + ? 50-59<br /> + ? - + +|+ + ? - - 60-69<br /> - ? + ? - - + - + ? 70-79<br /> - + - ? - + ? - - ? 80-89<br /> - - + + ? - - ? + - 90-99<br /> <br /> Now, if f(43) = 1, then f(61)=-1, and considering f[243,250] = -5-f(83), we have also that f(83)=-1. Also, f(47)=f(53)=f(59)=f(61)=-1, and the table would look like this:<br /> <br /> 0 1 2 3 4 5 6 7 8 9<br /> <br /> 0|+ + - + - - + + + 0-9<br /> - - - - + + + - + + 10-19<br /> - - - + - + - - + + 20-29<br /> + - + + - - + - + + 30-39<br /> - - -|+ - - + - - + 40-49<br /> + + - - - + + - + - 50-59<br /> + - - + +|+ + ? - - 60-69<br /> - ? + ? - - + - + ? 70-79<br /> - + - - - + + - - ? 80-89<br /> - - + + - - - ? + - 90-99<br /> <br /> The discrepancy up to 96 is now -7+f(67)+f(71)+f(73)+f(79)+f(89). Therefore f(67)=f(71)=f(73)=f(79)=f(89)=1. If we now say that f(107)=a, f(109)=b, f(163)=c, f(167)=d, f(331)=e. We have three choke points to consider here. First, f[155,172] = 5+f(157)+c+d, which implies that c+d&lt;=0. Second, f[213,218] = -4+a+b implies that a+b&gt;=0. Third, f[319,334]=-7-a-b+c+d+e. Plugging the results from the first two equations in, we get that f[319,334]&lt;=-7+e, which is impossible, since that would force the discrepancy to be greater than 3.<br /> <br /> Therefore f(43) = -1. The table is now:<br /> <br /> 0 1 2 3 4 5 6 7 8 9<br /> <br /> 0|+ + - + - - + + + 0-9<br /> - - - - + + + - + + 10-19<br /> - - - + - + - - + + 20-29<br /> + - + + - - + - + + 30-39<br /> - - -|- - - + ? - + 40-49<br /> + + - ? - + + - + ? 50-59<br /> + ? - + +|+ + ? - - 60-69<br /> - ? + ? - - + - + ? 70-79<br /> - + - ? - + - - - ? 80-89<br /> - - + + ? - - ? + - 90-99<br /> <br /> The discrepancy up to 48 is now -3-f(47), so f(47)=1. Therefore, f(53)=f(59)=f(61)=-1. But now, f[183,188]=6, which is a contradiction.<br /> <br /> Therefore f(29)=-1.</div>

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