In the equation 3x = 10 + a about X, a is a negative integer, and the solution of this equation is a natural number. Try to find the value of a and the solution of this equation

X = (10 + a) / 3 is a natural number, X may be 0,1,2,3
When a = - 10, x = 0;
When a = - 7, x = 1;
When a = - 4, x = 2;
When a = - 1, x = 3

The unknowns of quaternion Diophantine equation x ^ n + y ^ n = x ^ n + y ^ n are not equal to each other. It is speculated that there is a positive integer solution when n is only a few small natural numbers
n=1,2,3"

This is called the sum of equal powers problem. X and Y on the right side of the equation should be represented by different letters
For example, x ^ n + y ^ n = u ^ n + W ^ n
In fact, when n = 4, it has been proved that there are innumerable solutions, for example, the smallest one is found by Euler
133^4+134^4=158^4+59^4.

How to judge that n ^ 4 + n ^ 3-12n ^ 2-14n-16 = 0 has no solution?

Because
When n = 0, n ^ 4 + n ^ 3-12n ^ 2-14n-16 < 0;
When n = 10, n ^ 4 + n ^ 3-12n ^ 2-14n-16 > 0,
So the original equation has a real solution
There is no natural number solution!
Divide both sides of the equation by N & # 178
n²+n-(14n+16)/n²=12……………… ①
Because n & # 178; + N and 12 are natural numbers, so (14N + 16) / N & # 178; are natural numbers,
So, 14N + 16 ≥ n & # 178;
The solution is 7 - √ 65 ≤ n ≤ 7 + √ 65
So, n can be a natural number from 1 to 15,
Considering that 14N + 16 is even, n can only be even,
When n = 2,10, (14N + 16) / N & # 178; they are natural numbers 11 and 2 respectively, but they do not meet the requirements of (1),
Therefore, the original equation has no natural number solution

In the equation 3x = 10 + a about X, a is a negative integer, and the solution of this equation is a natural number. Try to find the value of a and the solution of this equation