20 Waring and Goldbach type problems, and Schnirelman’s constant
20.1 Waring Problem
Let \(A\subset \mathbf{N}\) be such that there exists \(k\) for which
Then \(A\) is called an additive basis of \(\mathbf{N}\). The minimum \(k\) for which 1 holds is the order of \(A\).
For any \(k\ge 1\) let \(A_k=\{ n^k:n\in \mathbf{N}\cup \{ 0\} \} \). Let \(g(k)\) be the order of \(A_k\) when it exists. That is, \(g(k)\) is the minimum number of \(k\) powers needed to write any natural number as the sum of (not necessarily unique) \(g(k)\) many \(k\) powers including 0.
For any \(k\ge 1\), let \(G(k)\) be the minimum \(m\) such that there exists \(N\ge 1\) for which
where \(J_N=\{ 1,\dots ,N\} \). That is, \(G(k)\) is the minimum number of \(k\) powers such that every sufficiently large integer may be written as the sum of (not necessarily unique) \(G(k)\) many \(k\) powers including 0.
20.1.1 Known values of g(k)
We have \(g(2)=4\); that is every natural number may be written as the sum of \(4\) perfect squares.
Linnik’s proof relied on the notion of Schnirelmann density, which will be discussed later.
In fact, the exact value of \(g(k)\) is known for almost all \(k\ge 1\). We have
and, otherwise,
where \(\xi =2\) if
and \(3\) otherwise. Note that \([x]\) is the greatest integer less than \(x\) and \(\{ x\} =x-[x]\). It has been shown that there at at most finitely many exceptions [ 202 ] . To complete the proof, it suffices to show
It has been shown for all \(k {\gt} 5000\), \(\{ (3/2)^k\} \le 1-a^{k}\) where \(a=2^{-0.9}\approx 0.53\), and for sufficiently large \(k\), \(\{ (3/2)^k\} \le 1-(0.5769\dots )^{-k}\) [ 61 ] [ 16 ] .
20.1.2 Known values of \(G(k)\)
Only 2 values of \(G(k)\) are known definitively: \(G(2)=4\) as shown by Lagrange and \(G(4)=16\) as shown by Davenport [ 53 ] .
Let \(G_1(k)\) be the smallest number \(m\) such that
where \(d(A)\) represents the natural density of \(A\):
\(G_1(k)\) has been determined for \(5\) values:
20.1.3 General bounds for \(G(k)\)
For all \(k\ge 1\),
Furthermore,
This bound is the sharpest to date and was a significant improvement over the previous bound by Wooley [ 299 ] : for sufficiently large \(k\),
20.1.4 Bounds for special cases for \(G(k)\)
\(k=3\)
\(G(3)\ge 4\)
Note cubes are congruent \(1,-1,0\) modulo \(9\). Thus, numbers congruent \(4,5\) modulo \(9\) may not be expressed as the sum of \(3\) cubes.
The exact value of \(G(3)\) is conjectured to be \(4\), but has not been proven.
Conjectured \(G(k)\) for small \(k\)
Table 20.2 summarizes the best upper bounds for \(G(k)\) and the conjectured values of \(G(k)\) for \(7\le k\le 20\).
|
7 |
8 |
9 |
10 |
11 |
12 |
13 |
14 |
15 |
16 |
17 |
18 |
19 |
20 |
|
|
Best Bound |
31 |
39 |
47 |
55 |
63 |
72 |
81 |
89 |
97 |
105 |
113 |
121 |
129 |
137 |
|
Conjectured |
8 |
32 |
13 |
12 |
12 |
16 |
14 |
15 |
16 |
64 |
18 |
27 |
20 |
25 |
The upper bounds for \(k\le 13\) were deduced from Wooley [ 300 ] and the bounds for \(14\le k\le 20\) are from Theorem 20.7.
20.1.5 Generalized Waring problem and connections to the Generalized Riemann Hypothesis
Waring’s Problem concerns the solvability of equations of the form
for \(m,n,k\in \mathbf{N}\), and Theorem 20.5 states that for any fixed \(k\ge 1\), there exists \(n\in \mathbf{N}\) such that (2) is solvable for all \(m\in \mathbf{N}\). A more generalized problem arises when \(k\) is not fixed. Given any \(\mathbf k=(k_1,\dots ,k_n)\in \mathbf{N}^n\), the generalized Waring problem concerns the solvability of the equations of the form
The following theorem due to Erich Kamke provides a partial result.
Let \(f(x)\) be an integer valued polynomial such that there does not exist \(d\in \mathbf{N}\) such that \(d|f(n)\) for all \(n\in \mathbf{N}\). Then for sufficiently large \(k\),
is solvable for all large enough \(m\).
Assuming the Generalized Riemann Hypothesis (GRH), The solvability of (3) can be guaranteed for specific \(\mathbf k\). For example:
Assuming GRH, then
is solvable for sufficiently high \(n\). Furthermore, (4) is not solvable for at most \(O((\log N)^{3+\epsilon })\) integers between \(1\) and \(N\).
20.2 Goldbach-Style Problems
Goldbach’s original conjecture stated that every positive integer could be written as the sum of \(3\) primes. In light of Waring’s problem, a natural extension of Goldbach’s problem asks when
is solvable for all \(n\in \mathbf{N}\) for \(p_1,\dots p_k\) prime and \(k\in \mathbf{N}\). It is conjectured when \(m\ge k+1\) and for sufficiently large \(n\) satisfying local conditions, which will be made more explicit for specific values of \(k\), (5) is solvable.
20.2.1 When \(k=2\)
It is conjectured that
is solvable whenever \(n\equiv 4\; (\operatorname {mod}\; 24)\). The following theorem gives the closest solution.
Let \(E(N)\) be the number of integers \(n\equiv 4\; (\operatorname {mod}\; 24)\) for which (6) is not solvable. For any \(\epsilon {\gt}0\),
20.2.2 When \(k=4,5\)
Following Kawada and Wooley, we define the following quantities to give the relevant local conditions for the cases \(k=4\) and \(k=5\). Let \(\theta =\theta (p,k)\) be the greatest power of \(p\) dividing \(k\); that is \(p^\theta |k\) but \(p^{\theta +1}\nmid k\). Then, let
and
In particular, \(K(4)=240\) and \(K(5)=2\).
For \(k\in \mathbf{N}\), let \(H(k)\) be the minimum integer \(s\) such that
is solvable for sufficiently large \(n\) whenever \(n\equiv s\; (\operatorname {mod}\; K(k))\).
Finding the value of \(H(k)\) is the main focus of the modern Waring-Goldbach problem.
We have
\(H(4)\le 14\)
For any positive \(A\),
\[ p_1^4+p_2^4+\cdots +p_7^4=n \]has at most \(O(N(\log N)^{-A})\) exceptions for \(n\equiv 7\; (\operatorname {mod}\; 240)\) and \(1\le n\le N\).
\(H(5)\le 21\)
For any positive \(A\),
\[ p_1^5+p_2^5+\cdots +p_{11}^5=n \]has at most \(O(N(\log N)^{-A})\) exceptions for \(n\) odd and \(1\le n\le N\).
In 2014, Zhao improved the bound for \(k=4\) and showed \(H(4)\le 13\) [ 316 ] . He also showed \(H(6)\le 32\) in the same paper.
20.2.3 When \(k\ge 7\)
For large values of \(k\),
Table 20.3 summarizes the best bounds on \(H(k)\) for \(7\le k\le 20\).
|
7 |
8 |
9 |
10 |
11 |
12 |
13 |
14 |
15 |
16 |
17 |
18 |
19 |
20 |
|
45 |
57 |
69 |
81 |
93 |
107 |
121 |
134 |
149 |
163 |
177 |
193 |
207 |
223 |
20.3 Schnirelmann Density
20.3.1 Existence of Additive Basis
Define the Schnirelmann density of \(A\subset \mathbf{N}\) as
Define the lower asymptotic density of \(A\subset \mathbf{N}\) as
Suppose \(\sigma A{\gt}0\). Then \(A\) is an additive basis for \(\mathbf{N}\).
With Theorem 20.18, it is possible to prove many sets form additive bases. For example:
Let \(\mathbb P\) denote the set of primes. Then, \(\delta (\mathbb P+\mathbb P){\gt}0\). Therefore, \(\mathbb P\) is an additive basis for \(\mathbf{N}\). The order of \(\mathbb P\) is denoted \(C\) and called Schnirelmann’s constant.
Schnirelmann originally bounded \(C{\lt}80000\) and Helfgott showed in 2013 that \(C\le 4\) [ 117 ] . Goldbach’s conjecture claims that \(C=3\).
Let \(\mathfrak S_a=\{ p+a^k:p\in \mathbb P, k\in \mathbf{N}\} \). Then, \(\sigma \mathfrak S_a{\gt}0\) for all \(a\in \mathbf{N}\). Thus, each integer \(n\) may be written as the sum of at most \(C_a\) primes and \(C_a\) powers of \(a\), where \(C_a\) is a constant depending only on \(a\).
20.3.2 Essential Components
\(B\subset \mathbf{N}\) is called an essential component if \(\sigma (A+B){\gt}\sigma (A)\) for any \(A\subset \mathbf{N}\) with \(0{\lt}\sigma A{\lt}1\).
Linnik showed in 1933 gave the first example of an essential component that is not a basis [ 187 ] . Erdos showed in 1936 that every basis is also an essential component [ 64 ] . The minimum possible size of an essential component remained an open problem until Ruzsa showed in 1984 [ 256 ] that for any \(\epsilon {\gt}0\), there exists an essential component \(H\) such that
but there does not exist an essential component such that