How many natural numbers m are such that: (2m-124)^3-=(3m+139)^3(mod(m+3))

Answer 1

Notice that for (3 m +139)³ mod (m+3) < m+3.

(2 m -124)³ grows unbounded as m goes toward infinity.

For a solution to exist

(2 m - 124)³ = (3 m +139)³ mod (m+3) < m+3.

(2 m - 124)³ < m+3. puts the condition 0 ≤ m ≤ 64.

Checking these numbers gives one solution m = 62.

I used this python code to check.

for n in range(65):

if (2*n-124)**3 == ((3*n+139)**3)%(n+3):

print(n)