Side Proof 10

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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(11)=f(17)=f(31)=1, f(7)=f(13)=f(23)=f(29)=-1.

Proof

It seems that we can't derive anything just from these assumptions.

Case 1: f(37)=1

Now, s(44) = 4+f(41)+f(43), so f(41)=f(43)=-1. We have two inequalities:

1) f[423,430] = 5-f(61)-f(71)+f(107) <= 4

2) s(74) = 5+f(59)+f(61)+f(67)+f(71)+f(73) <= 2

3) f[207,222] = -7-f(71)-f(73)+f(107)+f(109)+f(211) >= -4

(1)+(2)-(3)-17: f(59)+f(67)+f(71)+2f(73)-f(109)-f(211) <= -7, so f(59)=f(67)=f(71)=f(73)=-1, f(109)=f(211)=1.

We have another two inequalities:

1) f[141,160] = -6+f(79)+f(149)+f(151)+f(157) >= -4

2) f[471,476] = -4-f(79)-f(157) >= -4

(1)+(2)+10: f(149)+f(151) >= 2

Therefore, f(149)=f(151)=1

f[287,302] = 6-f(97)+f(293), so f(97)=1, f(293)=-1.

We have another two inequalities:

1) f[187,206] = 8+f(101)+f(103)+f(191)+f(193)+f(197)+f(199) <= 4

2) f[101,112] = -5+f(101)+f(103)+f(107) >= -4

(2)-(1)+13: f(107)-f(191)-f(193)-f(197)-f(199) >= 5, so f(107)=1, f(191)=f(193)=f(197)=f(199)=-1. However, now f[423,430] = 7-f(61), which forces the discrepancy above 3. Therefore, f(37)=-1.

Case 2: f(41)=1

Now, s(44) = 4+f(41)+f(43), so f(41)=f(43)=-1. We have two inequalities:

1) f[423,430] = 5-f(61)-f(71)+f(107) <= 4

2) s(72) = 4+f(59)+f(61)+f(67)+f(71) <= 2

(1)+(2)-9: f(59)+f(67)+f(107) <= -3. Therefore, f(59)=f(67)=f(107)=-1.

f[373,378] = 5+f(373), so f(373)=-1. f[205,210] = -5+f(103), so f(103)=1.

It seems this is as far as we can get with this assumption.

Case 2.1: f(41)=f(61)=1

Now, s(72)=3+f(71), so f(71)=-1. f[549,554] = 5+f(277), so f(277)=-1. f[635,640] = -5-f(127), so f(127)=-1.

It seems that this is again as far as we can get.

Case 2.1.1: f(41)=f(61)=f(73)=1

f[943,952] = -7-f(79)-f(317)+f(947), so f(79)=f(317)=-1, f(947)=1. f[141,160] = -7+f(149)+f(151)+f(157), so f(149)=f(151)=f(157)=1.

We have two inequalities:

1) f[309,320] = -4+f(311)+f(313) >= -4

2) f[621,628] = 5-f(89)+f(311)+f(313) <= 4

(1)-(2)+9: f(89) >= 1. Therefore, f(89)=1. s(100) = 4+f(83)+f(97), so f(83)=f(97)=-1.

We have another four inequalities:

1) f[161,178] = 5+f(163)+f(167)+f(173) <= 4

2) f[317,334] = -4-f(109)+f(163)+f(167)+f(331) >= -4

3) f[339,346] = -4-f(113)+f(173) >= 4

4) f[101,118] = -5+f(101)+f(109)+f(113) >= -4

(2)-(1)+(3)+(4)+18: f(101)+f(331) >= 2, so f(101)=f(331)=1. f[303,320] = -7+f(307)+f(311)+f(313), so f(307)=f(311)=f(317)=1. However, now f[621,628] = 6, forcing the discrepancy above 3. Therefore, f(73)=-1.

Case 2.1.2: f(41)=f(61)=1

s(80)=-3+f(79), so f(79)=1. f[471,476] = -5-f(157), so f(157)=-1. However, now f[141,160] = -8+f(149)+f(151), which is a contradiction. Therefore, f(61)=-1.

Case 2.2: f(41)=1

f[107,126] = -6+f(109)+f(113), so f(109)=f(113)=1. f[339,346] = -5+f(173), so f(173)=1. f[161,190] = 10+f(83)+f(89)+f(163)+f(167)+f(179)+f(181), so f(83)=f(89)=f(163)=f(167)=f(179)=f(181)=-1. We have two equations:

1) f[871,876] = 4-f(73)+f(97) <= 4

2) s(126) = -5+f(71)+f(73)+f(79)+f(97)+f(101) >= -2

(2)-(1)+7: f(71)+2f(73)+f(79)+f(101) >= 3, so f(73)=1.

f[207,220] = -6-f(71)+f(211), so f(71)=-1, f(211)=1. However, now f[423,430] = 6, which is impossible. Therefore, f(41)=-1.

Therefore, since s(56) = -5+f(43)+f(47)+f(53), f(43)=f(47)=f(53)=1.

Case 3: f(59)=1

s(72) = 5+f(61)+f(67)+f(71), so f(61)=f(67)=f(71)=-1. Now, f[423,428] = 5+f(107), so f(107)=-1. f[207,220] = -7-f(73)+f(109)+f(211), so f(73)=-1, f(109)=f(211)=1. s(80) = -3+f(79), so f(79)=1. It seems that we can't get much further with this assumption, so we have to add a few more assumptions.

Case 3.1: f(59)=f(83)=1

s(100)=4+f(89)+f(97), so f(89)=f(97)=-1. f[525,534] = 5+f(263), so f(263)=-1.

We have two equations:

1) f[165,190] = 7+f(163)+f(167)+f(173)+f(179)+f(181) <= 4

2) 7-f(179)-f(181)+f(269)+f(271)+f(541) <= 4

(1)+(2)-13: f(163)+f(167)+f(173)+f(269)+f(271)+f(541) <= -6

Therefore, f(163)=f(167)=f(173)=f(269)=f(271)=f(541)=-1. f[511,520] = 6-f(103)+f(257), so f(103)=1, f(257)=-1. However, now f[249,264] = -7+f(251), contradiction. Therefore, f(83)=-1.

Case 3.2: f(59)=f(89)=1, f(83)=-1

We have two equations:

1) f[161,190] = 7+f(163)+f(167)+f(173)+f(179)+f(181) <= 4

2) f[319,334] = -5+f(163)+f(167)+f(331) >= -4

(1)-(2)-12: f(173)+f(179)+f(181)-f(331) <= -4. Therefore, f(173)=f(179)=f(181)=-1, f(331)=1.

Now we have f[531,540] = 5+f(269), so f(269)=-1. Now, we have f[511,540] = 10-f(103)+f(131)+f(257)+f(263)+f(521)+f(523) <= 4, so f(103)=1, f(131)=f(257)=f(263)=f(521)=f(523)=-1. However, now f[261,268] = -6, which is a contradiction. Therefore, f(89)=-1.

Case 3.3: f(59)=1, f(83)=f(89)=-1

We have two equations:

1) s(108) = -3+f(97)+f(101)+f(103) >= -2

2) f[187,206] = 7+f(97)+f(101)+f(103)+f(191)+f(193)+f(197)+f(199) <= 4

(2)-(1)-10: f(191)+f(193)+f(197)+f(199) <= -4, so f(191)=f(193)=f(197)=f(199)=-1.

f[511,540] = 6-f(179)+f(269), so f(179)=1, f(269)=-1.

We have a two equations:

1) f[161,190] = 6+f(163)+f(167)+f(173)+f(181) <= 4

2) f[319,334] = -5+f(163)+f(167)+f(331) >= -4

(1)-(2)-11: f(173)+f(181)-f(331) <= -3, so f(173)=f(181)=-1, f(331)=1.

Now, f[511,544] = 12-f(103)+f(131)+f(257)+f(269)+f(271)+f(521)+f(523)+f(541) <= 4, so f(103)=1, f(131)=f(257)=f(269)=f(271)=f(521)=f(523)=f(541)=-1.

We also have that f[261,274] = -5+f(137), so f(137)=1. But now f[135,140] = 6, which is a contradiction. Therefore, f(59)=-1.

Case 4: f(61)=1

s(72) = 4+f(67)+f(71), so f(67)=f(71)=-1.

It seems we cannot derive anything more from here, so we have to make a new assumption

Case 4.1: f(61)=f(73)=1

Now, f[207,220]=-7+f(107)+f(109)+f(211), so f(107)=f(109)=f(211)=1. Also, f[287,292] = 5-f(97), so f(97)=1. However, now s(100)=5+f(79)+f(83)+f(89), so f(79)=f(83)=f(89)=-1. f[319,334] = -7+f(163)+f(167)+f(331), so f(163)=f(167)=f(331)=1. f[161,190] = 7+f(173)+f(179)+f(181), so f(173)=f(179)=f(181)=-1.

We have two inequalities:

1) f[187,202] = 8+f(101)+f(103)+f(191)+f(193)+f(197)+f(199) <= 4

2) f[573,586] = 6-f(191)-f(193)+f(293)+f(577) <= 4

(1)+(2)-14: f(101)+f(103)+f(197)+f(199)+f(293)+f(577) <= -6, so f(101)=f(103)=f(197)=f(199)=f(293)=f(577)=-1.

s(118) = -5+f(113), so f(113)=1.

We have another two inequalities:

1) f[293,328] = -10+f(149)+f(151)+f(157)+f(163)+f(307)+f(311)+f(313)+f(317) >= -4

2) f[621,628] = 5+f(151)+f(311)+f(313) <= 4

(1)-(2)+15: f(149)+f(157)+f(163)+f(307)+f(317) >= 7, which is impossible. Therefore, f(73)=-1.

Case 4.2: f(61)=1, f(73)=-1

s(80)=-5+f(79), so f(79)=1.