Human proof that completely multiplicative sequences have discrepancy greater than 2

It is known from computer search that the longest completely multiplicative sequence [math]\displaystyle{ f: [N] \to \{-1,+1\} }[/math] of discrepancy 2 has length 246. Here we record the attempts to prove this by hand. Notation: we display our function in rows of 10, starting with 0. We use the notation that + and - denote values which are known to equal +1 or -1 respectively, and ? for unknown values. Thus, initially our array looks like this:

0 1 2 3 4 5 6 7 8 9

0|+ ? ? + ? ? ? ? +   0-9
? ? ? ? ? ? + ? ? ?   10-19
? ? ? ? ? + ? ? ? ?   20-29
? ? ? ? ? ? + ? ? ?   30-39

since f(n^2) = 1 for all square numbers.

The notions of a cut and a block will be useful. A cut occurs between n and n+1 whenever it can be proved that f[1,n] sums to 0; these can only occur when n is even. We indicate a cut by a bar | between n and n+1. Thus for instance we begin with just one cut:

0 1 2 3 4 5 6 7 8 9

0|+ ? ? + ? ? ? ? +   0-9
? ? ? ? ? ? + ? ? ?   10-19
? ? ? ? ? + ? ? ? ?   20-29
? ? ? ? ? ? + ? ? ?   30-39

A block is any interval between two cuts; thus inside each block one must sum to zero, and the discrepancy inside each block is at most 2. Thus we can "localise" the analysis to each block, though multiplicativity relates the blocks together.

A block is solved if all values are assigned in it. Solved blocks are uninteresting (other than for the values they provide by multiplicativity), so we shall adopt the convention of concatenating solved blocks with adjacent blocks to reduce clutter.

More notation: we use f[a,b] as short for f(a)+...+f(b). Note that

f[a,b] = -4,-2,0,2,4 if a is odd and b is even;

f[a,b] = -3,-1,1,3 if a and b are both odd or both even;

f[a,b] = -2,0,2 if a is even and b is odd.

Also if [a,b] starts or ends at a cut, then f[a,b] has magnitude at most 2.

A key observation is that if f(2n)=f(2n+1) then there must be a cut between 2n and 2n+1.

Case 1. f(2)=1

Now, let us assume f(2)=1 and that the discrepancy is at most 2 and see what happens. From multiplicativity we obtain f(2n^2)=1, thus

0 1 2 3 4 5 6 7 8 9

0|+ + ? + ? ? ? + +   0-9
? ? ? ? ? ? + ? + ?   10-19
? ? ? ? ? + ? ? ? ?   20-29
? ? + ? ? ? + ? ? ?   30-39

As f(8)=f(9) we can now cut at 8=9. And then by using discrepancy we get f(3)=f(5)=-1, thus

0 1 2 3 4 5 6 7 8 9

0|+ + - + - ? ? +|+   0-9
? ? ? ? ? ? + ? + ?   10-19
? ? ? ? ? + ? ? ? ?   20-29
? ? + ? ? ? + ? ? ?   30-39

which then by multiplicativity and adding cuts where obvious extends to

0 1 2 3 4 5 6 7 8 9

0 + + - + - -|? +|+   0-9
-|? - ? ? + + ? + ?   10-19
- ? ? ? - + ? - ? ?   20-29
+ ? + ? ? ? + ? ? ?   30-39

Looking at the 7-8 block we see that f(7) is forced to be -1; from this and multiplicativity, and showing rows up to 100, we arrive at

0 1 2 3 4 5 6 7 8 9

0 + + - + - - - + +   0-9
-|? - ? - + + ? + ?   10-19
- + ? ? - + ? - - ?   20-29
+ ? + ? ? + + ? ? ?   30-39
- ? + ? ? - ? ? - +   40-49
+ ? ? ? - ? - ? ? ?   50-59
+ ? ? - + ? ? ? ? ?   60-69
+ ? + ? ? - ? ? ? ?   70-79
- + ? ? ? ? ? ? ? ?   80-89
- ? ? ? ? ? - ? + ?   90-99
+

Note that

f[15,19] = 3 + f(17) + f(19)

cannot exceed 3, and

f[168,171] = 2 - f(17) + f(19)

cannot exceed 2. Thus f(19) = -1. We use this information to update the board:

0 1 2 3 4 5 6 7 8 9

0 + + - + - - - + +   0-9
-|? - ? - + + ? + -   10-19
- + ? ? - + ? - - ?   20-29
+ ? + ? ? + + ? - ?   30-39
- ? + ? ? - ? ? - +   40-49
+ ? ? ? - ? - + ? ?   50-59
+ ? ? - + ? ? ? ? ?   60-69
+ ? + ? ? - - ? ? ?   70-79
- + ? ? ? ? ? ? ? ?   80-89
- ? ? ? ? + - ? + ?   90-99
+

Let's write f(11)=a, f(13)=b, f(17)=c, f(23)=e, f(29)=f and A = -a, B = -b, C = -c, E=-e, F=-f. Then we can fill in more:

0 1 2 3 4 5 6 7 8 9

0 + + - + - - - + +   0-9
-|a - b - + + c + -   10-19
- + a e - + b - - f   20-29
+ ? + A c + + ? - B   30-39
- ? + ? a - e ? - +   40-49
+ C b ? - A - + f ?   50-59
+ ? ? - + B A ? c E   60-69
+ ? + ? ? - - A B ?   70-79
- + ? ? + C ? F a ?   80-89
- B C ? ? + - ? + a   90-99
+

This seems to be about as far as one can get just from the assumption f(2)=+1.

Case 1.a f(2)=f(37)=1

The most profitable assumption here seems to be f(37)=+1. This creates a cut at 36-37, which then forces f(34)=-1 and then f(33)=-1 also. Filling in these values and adding in cuts, and extending a row gives

0 1 2 3 4 5 6 7 8 9

0 + + - + - - - + +   0-9
- + -|b - + + - + -   10-19
- + + e - + b - - f   20-29
+ ? +|- - + + + -|B   30-39
- ? + ? + - e ? - +   40-49
+|+ b ? -|- - + f ?   50-59
+ ? ? - + B - ? - E   60-69
+ ? + ? + - -|- B ?   70-79
- + ? ? +|+ ? F + ?   80-89
- B e ? ? + - ? +|+   90-99
+ ? + ? b - ? ? - ?   100-109

looking at f[30,32] we thus have f(31)=-1, while from f[99,103] we have f(101)=f(103)=-1,

0 1 2 3 4 5 6 7 8 9

0 + + - + - - - + +   0-9
- + -|b - + + - + -   10-19
- + + e - + b - - f   20-29
+|- + - - + + + -|B   30-39
- ? + ? + - e ? - +   40-49
+|+ b ? -|- - + f ?   50-59
+ ? - - + B - ? - E   60-69
+ ? + ? + - -|- B ?   70-79
- + ? ? +|+ ? F + ?   80-89
- B e + ? + - ? +|+   90-99
+ - + - b - ? ? - ?   100-109

From the [51,54] block we have f(53) = B, which then forces f[90,106]=0, so there is a cut at 106-107:

0 1 2 3 4 5 6 7 8 9

0 + + - + - - - + +   0-9
- + -|b - + + - + -   10-19
- + + e - + b - - f   20-29
+|- + - - + + + -|B   30-39
- ? + ? + - e ? - +   40-49
+|+ b B -|- - + f ?   50-59
+ ? - - + B - ? - E   60-69
+ ? + ? + - -|- B ?   70-79
- + ? ? +|+ ? F + ?   80-89
- B e + ? + - ? +|+   90-99
+ - + - b - B|? - ?   100-109
-|- - ? + E f b ? +   110-119
+|+ ? ? -|- - ? + ?   120-129

From f[118-120] we have f(59)=-1; from f[111,113] we have f(113)=+1;

0 1 2 3 4 5 6 7 8 9

0 + + - + - - - + +   0-9
- + -|b - + + - + -   10-19
- + + e - + b - - f   20-29
+|- + - - + + + -|B   30-39
- ? + ? + - e ? - +   40-49
+|+ b B -|- - + f -   50-59
+ ? - - + B - ? - E   60-69
+ ? + ? + - -|- B ?   70-79
- + ? ? +|+ ? F + ?   80-89
- B e + ? + - ? +|+   90-99
+ - + - b - B|? - ?   100-109
-|- - + +|E f b - +   110-119
+|+ ? ? -|- - ? + ?   120-129

From f[13,30] we have 2b+e+f=0 while from f[115,120] we have -e+f+b=-1. This forces b=-1, e=f=1:

0 + + - + - - - + +   0-9
- + - - - + + - + -   10-19
- + + + - + - - - +   20-29
+ - + - - + + + - +   30-39
-|? + ? + - + ? - +   40-49
+|+ - + - - - + + -   50-59
+|? -|- +|+ - ? -|-   60-69
+|? + ? + - -|- +|?   70-79
- + ? ? +|+ ? - + ?   80-89
- + +|+ ? + - ? +|+   90-99
+ - + - - - +|? - ?   100-109
-|- - + + - + - - +   110-119
+|+ ? ? -|- - ? + ?   120-129

From the 60-69 row we then get f(61)=+1, f(67)=+1; from f[85,92] we get f(43)=f(89)=-1; from f[93,98] we have f(47)=f(97)=-1; from f[125,127] one has f(127)=+1; from f[107,110] one gets f(107)=f(109)=+1:

0 + + - + - - - + +   0-9
- + - - - + + - + -   10-19
- + + + - + - - - +   20-29
+ - + - - + + + - +   30-39
-|? + - + - + - - +   40-49
+|+ - + - - - + + -   50-59
+ + - - + + - + -|-   60-69
+|? + ? + - -|- +|?   70-79
- + ? ? +|+ - - + -   80-89
- + + + - + - - + +   90-99
+ - + - - - + + - +   100-109
-|- - + + - + - - +   110-119
+|+ + ? -|- - + + ?   120-129

From f[41,50] one has f(41)=-1; but then f[121,125]=2, a contradiction. Hence this case is not possible.

Case 1.b f(2)=1, f(37)=-1

This takes us back to case 1.b f(2)=1, f(37)=-1, which begins here:

Let's write f(11)=a, f(13)=b, f(17)=c, f(23)=e, f(29)=f and A = -a, B = -b, C = -c, E=-e, F=-f. Then we can fill in more:

0 1 2 3 4 5 6 7 8 9

0 + + - + - - - + +   0-9
-|a - b - + + c + -   10-19
- + a e - + b - - f   20-29
+ ? + A c + + ? - B   30-39
- ? + ? a - e ? - +   40-49
+ C b ? - A - + f ?   50-59
+ ? ? - + B A ? c E   60-69
+ ? + ? ? - - A B ?   70-79
- + ? ? + C ? F a ?   80-89
- B C ? ? + - ? + a   90-99
+

So we assume f(37)=-1

0 1 2 3 4 5 6 7 8 9

0 + + - + - - - + +   0-9
-|a - b - + + c + -   10-19
- + a e - + b - - f   20-29
+ ? + A c + + - - B   30-39
- ? + ? a - e ? - +   40-49
+ C b ? - A - + f ?   50-59
+ ? ? - + B A ? c E   60-69
+ ? + ? -|- - A B ?   70-79
- + ? ? + C ? F a ?   80-89
- B C ? ? + - ? + a   90-99
-

From f[75,77] we have A=+1, then from f[11,13] we have b=-1, and then from f[75,79] we have f(79)=+1:

0 1 2 3 4 5 6 7 8 9

0 + + - + - - - + +   0-9
-|- - + - + +|c + -   10-19
- + - e - + + - - f   20-29
+ ? +|+ c|+ + - -|-   30-39
- ? + ? - - e ? - +   40-49
+ C + ? - + - + f ?   50-59
+ ? ? - + - + ? c E   60-69
+ ? + + - - - + - +   70-79
- + ? ? + C ? F - ?   80-89
- - C ? ? + - ? + -   90-99
+

From f[33,34] we have c=-1, and from f[39,41] we have f(41)=+1;

0 1 2 3 4 5 6 7 8 9

0 + + - + - - - + +   0-9
-|- - + - + +|- +|-   10-19
- + - e - + + - - f   20-29
+ ? + + -|+ + - - -   30-39
- + + ? - - e ? - +   40-49
+ + + ? - + - + f ?   50-59
+ ? ? - + - + ? + E   60-69
+ ? + + - - - + - +   70-79
- + ? ? + + ? F - ?   80-89
- - + ? ? + - ? + -   90-99
+

From f[19,23] we have e = +

0 1 2 3 4 5 6 7 8 9

0 + + - + - - - + +   0-9
-|- - + - + +|- +|-   10-19
- + - + - + +|- - f   20-29
+ ? + + -|+ + - - -   30-39
- + + ? - - + ? - +   40-49
+ + + ? - + - + f ?   50-59
+ ? ? - + - + ? + -   60-69
+ ? + + - - - + - +   70-79
- + ? ? + + ? F - ?   80-89
- - + ? ? + - ? + -   90-99
+

From f[27,29] we have f = +, so from f[27,34] we have f(31) = -, and sums and multiplicity force

0 1 2 3 4 5 6 7 8 9

0 + + - + - - - + +   0-9
-|- - + - + +|- +|-   10-19
- + - + - + +|- - +   20-29
+|- + + -|+ + - -|-   30-39
- + +|+ -|- +|- - +   40-49
+ + + - -|+ -|+ + -   50-59
+ - -|- + - +|? + -   60-69
+ ? + + - - - + - +   70-79
- + + ? + + + - - ?   80-89
- - + + - + - ? + -   90-99
+

The smallest sum before 70 is -1, so 71 must be -1. This forces 67 = -1, otherwise the sum at 73 is 3.

0 1 2 3 4 5 6 7 8 9

0 + + - + - - - + +   0-9
-|- - + - + +|- +|-   10-19
- + - + - + +|- - +   20-29
+|- + + -|+ + - -|-   30-39
- + +|+ -|- +|- - +   40-49
+ + + - -|+ -|+ + -   50-59
+ - -|- + - +|- +|-   60-69
+|- +|+ -|- - + - +   70-79
- + +|? + + + - - ?   80-89
- - + + - + - ? + -   90-99
+

83 must be -, otherwise the sum is 3 at 85.

0 1 2 3 4 5 6 7 8 9

0 + + - + - - - + +   0-9
-|- - + - + +|- +|-   10-19
- + - + - + +|- - +   20-29
+|- + + -|+ + - -|-   30-39
- + +|+ -|- +|- - +   40-49
+ + + - -|+ -|+ + -   50-59
+ - -|- + - +|- +|-   60-69
+|- +|+ -|- - + - +   70-79
- + +|- +|+ + - -|?   80-89
- - + + - + - ? + -   90-99
+

So 89 is + (otherwise the sum is -3 at 91)

0 1 2 3 4 5 6 7 8 9

0 + + - + - - - + +   0-9
-|- - + - + +|- +|-   10-19
- + - + - + +|- - +   20-29
+|- + + -|+ + - -|-   30-39
- + +|+ -|- +|- - +   40-49
+ + + - -|+ -|+ + -   50-59
+ - -|- + - +|- +|-   60-69
+|- +|+ -|- - + - +   70-79
- + +|- +|+ + - -|+   80-89
- - +|+ -|+ -|? + -   90-99
-

(could use intermediate diagrams displayed here with text)

f(58) is positive so f(116) is positive

f(13) is positive and f(9) is a square so f(117) is positive

there is a cut between 116 and 117

f(118) is negative since f(59) is negative

f(7) and f(17) are negative so f(119) is positive

f(3) and f(5) are negative and f(2) is positive so 120 is positive

121 is a square so f(121) is positive

so at 116 the sum is zero at because there is a cut between 116 and 117 at 117 the sum is one since f(117) is positive at 118 the sum is zero since f(118) is negative and since f(119) f(120) and f(121) are positive at 121 the sum is three and we have shown that if a completely multiplicative sequnce has f(2) equal to 1 and f(37) equal to -1 then it reaches discrepancy 3 before 246.

Since we have previously shown that if a completely multiplicative sequence that has f(2) equal to 1 and f(37)equal to 1 reaches discrepancy 3 before 246 we have any completely multiplicative sequence with f(2) equal to one reaches discrepancy 3 before 246 and that finishes case 1.

Case 2. f(2) = -1

Therefore f(2) = -1. Resetting the board:

0 1 2 3 4 5 6 7 8 9

0 + - ? + ? ? ? ? +   0-9
? ? ? ? ? ? + ? ? ?   10-19
? ? ? ? ? + ? ? ? ?   20-29
? ? ? ? ? ? + ? ? ?   30-39
? ? ? ? ? ? ? ? ? +   40-49
? ? ? ? ? ? ? ? ? ?   50-59
? ? ? ? + ? ? ? ? ?   60-69
? ? ? ? ? ? ? ? ? ?   70-79
? + ? ? ? ? ? ? ? ?   80-89
? ? ? ? ? ? ? ? ? ?   90-99

Here is a computer proof that completely multiplicative sequences have at least 2, inspired by the above analysis.