Goldbach
In
Unit 18, at the end, you look at the Taylor series
f(x) = ∑
p x
p = x
2 + x
3 + x
5 + ....
This function encodes all the primes. Now, the Taylor series of f(x)
2 tells in how
many ways a number can be written as a sum of two primes. So, in order to see these numbers,
we form the function f(x) by taking say up to say n=40:
f(x) = x
2 + x
3 + x
5 + x
7 + x
11 + x
13
+ x
17 + x
19 + x
23 + x
29 + x
31 + x
37
then form g(x) = f(x)
2 to get
g(x) = x
4 + 2 x
5 + x
6 + 2 x
7 + 2 x
8 + 2 x
9 + 3 x
10 + 2 x
12 +
2 x
13 + 3 x
14 + 2 x
15 + 4 x
16 + 4 x
18 + 2 x
19 + 4 x
20 + ...
We see for example that 10 can be written in three ways as a sum of two primes: 3+7, 7+3, 5+5. This number can be obtained by
taking the 10th derivative of g and divide by 10! The Goldbach conjecture asks that all even derivatives of g are positive. It is still
unsolved. If you want to experiment, here is the Mathematica code which gives the numbers. We take n=400 which allows us to see all the
terms up to n/2 = 200. If we plot this, we see what one calls the
Goldbach comet.
n=400; f=Sum[If[PrimeQ[a],x^a, 0], {a,n}]; g=Expand[f*f];
G=CoefficientList[g,x]; s=Table[G[[2 k-1]],{k,3,n/2}];
ListPlot[Flatten[s],AspectRatio->1,Joined->True]
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The pictures show the comet for n/2=200 and n/2=400. |
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Are there a multivariable versions? Yes, one can look at Gaussian primes
which are Gaussian integers a+i b. The analog claim is that if we look at
f(x,y) = ∑
p=a+ib,a>0,b>0 x
a y
b, then all
the Taylor entries a
n,m of g=f*f are positive if n+m is even, bigger than 2.
This is the Gaussian Goldbach conjecture. See
this project
and
Paper. We know that the conjecture
is hard to prove if it is true as it implies as a special case that there are infinitely
many rational primes of the form n
2 + 1, which is the Landau conjecture, which is
widely considered to be a difficult problem. In any case, the problem is also a problem
in calculus, because we are interested in the Taylor expansion of a concrete function g(x,y)
of two variables. Like in this
story told
at the occasion of the 300th birthday of Euler, one can try to find some positive
constants c(n,m) such that f(x,y) = ∑
p=a+ib,a>0,b>0 c(a,b) x
a y
b
is expressible using known functions like exp, sin, cos, etc, then just compute the derivatives of
f(x,y)^2 and establish so the conjecture. Unfortunately, no such choice of coefficients have been
found (and in the classical Goldbach case, one has tried since more than 100 years).
n=20; f=Sum[If[PrimeQ[a+I b,GaussianIntegers->True],x^a y^b,0],{a,n},{b,n}];
g=Expand[f*f]; G = CoefficientList[Expand[f*f],{x,y}];
s=Table[G[[i,j]],{i,3,n/2},{j,3,n/2}]
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Here is the start of the function
f(x,y) = x*y + x
2*y + x
4*y + x*y
2 + x
3*y
2 + x
5*y
2 + x
2*y
3 + x*y
4 + x
5*y
4 + x
2*y
5 + x
4*y
5 ...
The function g(x,y) = f(x,y)
2 now starts as
g(x,y) = x^2*y^2 + 2*x^3*y^2 + x^4*y^2 + 2*x^5*y^2 + 2*x^6*y^2 + x^8*y^2 + 2*x^2*y^3 + 2*x^3*y^3 + ...
We see the coefficients in the following Goldbach matrix.