Any results for small number Goldbach conjecture research?

Question

It seems to me that most research results on Goldbach conjecture research are for large number. (Example: results of Vinogradov, Terence Tao, Harald Helfgott, etc).

My understanding is that those results essentially depend on 1. there are lots of primes 2. "probability argument"

Any results for small number Goldbach conjecture research ?

For example, to prove (or to explain) why Goldbach conjecture hold true for number < 1,000 ?

(This is almost like a high school teacher ask students to prove Goldbach conjecture for number < 1,000 without relying on numerical verification for every number).

Answer 1

To explain why Goldbach's conjectures hold true for the number $<1000$ you can do it this way:

case I

$2n= 0 \bmod 6$

$2n=(-1+6a)+(+1+6b)=p_a+p_b$

$p_a= -1 \bmod 6,p_b= 1 \bmod 6$

Example $2n=96$

In this graph the pairs of numbers $2n=p_a+p_b$ with $p_a= -1 \bmod 6$ and $p_b= 1 \bmod 6$

It can be verified that for $ 2n <1000 $ between two consecutive prime $ p_b = 1 \bmod 6 $ there are at most $ 4 $ composite numbers $ 1 \bmod 6 $ (example between the prime 43 and prime 61 there are the two composite numbers 49 and 55)

for this reason for at least one of these values ​​$ {5,11,17,23,29} $ for $ p_a $ (note that there are $5$ consecutive $-1 \bmod 6 $ prime numbers) there exists a prime number $ p_b = 2n- p_a $

case II

$2n= -2 \bmod 6$

$2n=(-1+6a)+(-1+6b)=p_a+p_b$

$p_a,p_b= -1 \bmod 6$

and

$2n= -2 \bmod 6= 4 \bmod 6=3+(+1+6c)=3+p_c$

$p_c=2n-3= 1 \bmod 6$

Example $2n=94$

In this graph the pairs of numbers $2n=p_a+p_b$

$p_a,p_b= -1 \bmod 6$

Note the symmetry with respect to $n$.

and

$2n= 3+ 91$

$p_c=91= 1 \bmod 6$

If we don't consider the cases in which $ p_c = 2n-3 $ are prime. It can be verified that for $ 2n \leq 1000 $ excluding the case $ p_b = 557 $ between two consecutive prime $ p_b = -1 \bmod 6 $ there are at most $ 4 $ composite numbers $ -1 \bmod 6 $

for this reason for at least one of these values ​​$ {5,11,17,23,29} $ for $ p_a $ there exists a prime number $ p_b = 2n- p_a $. For excluded cases we have $ 2n = 556 = 509 + 47 $.

case III

$2n= 2 \bmod 6$

$2n=(+1+6a)+(+1+6b)=p_a+p_b$

$p_a,p_b= 1 \bmod 6$

and

$2n=3+(-1+6c)=3+p_c$

$p_c=2n-3 =-1 \bmod 6$

Example $2n=98$

In this graph the pairs of numbers $2n=p_a+p_b$

$p_a,p_b= 1 \bmod 6$

Note the symmetry with respect to $n$.

and

$2n= 3+ 95$

$p_c=95= -1 \bmod 6$

If we don't consider the cases in which $ p_c = 2n-3 $ are prime. It can be verified that for $ 2n <68 $ between two consecutive prime $ p_b = 1 \bmod 6 $ there are at most $ 2 $ composite numbers $ 1 \bmod 6 $

for this reason for at least one of these values ​​$ {7,13,19} $ for $ p_a $ there exists a prime number $ p_b = 2n- p_a $

and that for $ 68 \leq 2n <1000 $ excluding the case $ p_b = 271 $ and $ p_b = 967 $ between two consecutive prime $ p_b = 1 \bmod 6 $ there are at most $ 3 $ composite numbers $ 1 \bmod 6 $

for this reason for at least one of these values ​​$ {61,67,73,79} $ for $ p_a $ there exists a prime number $ p_b = 2n- p_a $ . For excluded cases we have $ 2n = 326 = 313 + 13 $ .

Answer 2

If you don't mind Excel, try this illustrative app, which shows exactly why the conjecture holds true for small numbers:

The distribution of primes in Goldbach partitions are described by two linear equations: x-intercept lines (red) and y-intercept slopes (blue).

This app provides the ability to experiment with hypothetical gaps. http://michaelmross.com/GoldbachStressTest1.xlsm
(You must enable editing and enable macros to use.)