fresh_42 said:
Looks o.k. in general. Just a few remarks. I think there are some further steps needed.
It is clear that all ##x_i \cdot y_j ## divide ##x \cdot y##. But why does every divisor have this form?
Given ##d\,|\,(xy)##, show that ##d=x_iy_j## for some pair ##(i,j)\,?##
The problem is the following: If ##d|(xy)## then ##d=x_{i_1}\ldots x_{i_k}y_{j_1}\ldots y_{j_l}##, i.e. the divisors of ##d## are somehow distributed over the divisors of ##x## and ##y##. But we cannot conclude that ##x_{i_1}\ldots x_{i_k}## is again a divisor of ##x##, e.g. ##2|12## and ##4|12## but ##8\nmid 12\,.## So how does ##d## have the desired form?
In other words: ##s(2)\cdot s(6) = 3 \cdot 12 = 36 \neq 28 = s(12)##. So ##s(xy)\neq s(x)s(y)## in general. However, with an additional assumption it is true, and you only need it in the restricted version. Which is the additional assumption and can you prove it in that case?
Hmmm... s(2)=1+2=3, s(3)=1+3=4, s(2*3)=s(6)=1+2+3+6=12=s(2)*s(3)... in the same way s(2)=1+2=3, s(4)=1+2+4=7, s(2*4)=s(8)=1+2+4+8=15... holy choo-choo, that's different than s(2)*s(4)! I forgot something! Thanks Lord that they don't let high-schoolers design homes, otherwise they would forget something like the doors and everybody would have to enter through the roof!
I suspect s(a*b)=s(a)*s(b) only if a and b have no common divisors other than 1 (no common factors). How to prove... for s(2) and s(4), the factors are {1,2} and {1,2,4}, therefore the ##x_i*y_j## forms would be {1*1, 1*2, 1*4, 2*1, 2*2, 2*4} = {1,2,4,2,4,8}... so the 2 and 4 are counted twice... that's because ##x_i1*y_j1## = ##x_i2*y_i2## for some i1 <> i2. If we separate these double-counted forms, then s(2)*(4)-(2+4)=(1+2+4+8)=15=s(8), so s(2*4) = s(2)*s(4) - M. So I suspect in general s(a*b) = s(a)*s(b) - M.
So, two troubles: how to prove M=0 if a and b have no common factors, and how to prove that if d|(x*y) then ##d=x_i*y_j##. Hmm... let's try this without cheating (I cheated too much already by spying on Euler's!).
Defining ##d|z## is true if d is a divisor of z.
I) if d|(x*y) then d is a product of a factor of x and a factor of y; this can be proved by logical arguments (not sure how to prove it an algebraically correct way), as:
expressing z=x*y,d,x,y as the product of primes (notice these are different than ##x_i, y_i## and ##z_i## from the previous post!: ##z=2^{z_2}*3^{z_3}*5^{z_5}*..., d=2^{d_2}*3^{d_3}*5^{d_5}*..., x=2^{x_2}*3^{x_3}*5^{x_5}, y=2^{y_2}*3^{y_3}*5^{y_5}##, then for every prime factor p, ##z_p=x_p+y_p##
as ##d|z##, ##d_p<=z_p##, ie ##d_p<=x_p+y_p##; define:
##dx_p = min( x_p, d_p )##, what makes ##p^{dx_p}|x##
##dy_p = d_p - dx_p##, what makes ##p^{dy_p}|y## [Note]
[Note: ##d_p <= x_p+y_p##, if ##dx_p=d_p## then ##dy_p=0## ie ##p^{dy_p}|y##; otherwise if ##dx_p=x_p## that means ##d_p - dx_p <= x_p+y_p - x_p = y_p##, or ##dy_p - dx_p <= y_p## therefore ##p^{dy_p}|y##]
then name
## u=2^{dx_2}*3^{dx_3}*...##, that makes ##u|d## and ##u|x##
## v=2^{dy_2}*3^{dy_3}*...##, that means ##v|d## and ##v|y##
we get d=u*v for the reason that ##dx_p+dy_p=d_p##
so, if ##d|(x*y)##, then there exists two numbers u and v, such that ##d=u*v## and ##u|x## and ##v|z##. This holds true even if they have common factors.