Why is no perfect square a multiple of 3n+2?
The statement= n^2= x(3n+2). Where x is any number from 1-below infinity. (I will not count infinity as a solution since technically infinity does not have a numerical perfect square)
This is because no number squared can give 3n+2. If you try 1 and plug it into the function, it will give you 3+2 = 5. If you try 2, it will give you 8. 4 is not a multiple of 8. Using higher prime numbers: 123^2= 15129 plugged into 3n+2 = 371. 15129/371=40.7789757412. (Most equations use a calculator (hope you dont mind)). Also, using the largest 5 digit prime number = 99991. 99991^2=9998200081. plugged into 3n+2=299975 9998200081/299975=33330.1111126. Which is not a whole number.
After plotting into desmos, we can only see ONLY 2 points, (0,0) and (-0.889, -1.333). Using a calculator, we can calculate that the slope is 3.0002001133476 which is unfortunately is not a whole number. This means that no matter what you input as n, you will never get a whole number.
(I have no idea if this is right. Just a stab in the dark)