Firstly, the conjecture "If T is a triangular number, 8T+1 is a square number." is proven by using the general rule that the sequence of triangle numbers is n(n+1)/2. However, before I continue with the proof, I will show a proof for why is this so (source: https://www.mathsisfun.com/algebra/triangular-numbers.html)
To do this we must create a pattern of dots, 1 dot, 3 dots, 6 dots, and so on in the shape of triangles (shown in image). Now, double the number of dots to make a rectangle. One pair of sides of the rectangle will be of length n (which is the sequence number too, pattern 1 originally has 1 dot, 2 has 3, 3 has 6 etc.) and the other pair of sides will be (n+1). So the area of the rectangle is n(n+1). Half the area will give the area of the triangle. therefore the number of dots is n(n+1)/2. And that is the nth term of the sequence.
Or alternatively, we could find the quadratic sequence of the numbers
Sequence: 1 , 3 , 6 , 10 , 15 , .....
Difference: 2 , 3 , 4 , 5 , .....
2nd level diff.: 1 , 1 , 1 , ...
Therefore first term of nth term = (1/2!)n^2 = (n^2)/2
Now we find the rest of the sequence, which will be linear
n: 1 , 2 , 3 , 4 , 5
Sequence: 1 , 3 , 6 , 10 , 15 , ....
(n^2)/2: 1/2, 2 , 9/2 , 8 , 25/2, ....
(n^2)/2-sequence: 1/2, 1 , 3/2 , 2 , 5/2
nth term of (n^2)/2 - sequence = n/2
Therefore, overall sequence = (n^2)/2 + n/2 = n(n+1)/2
Now that the general rule is proven we can carry on with the proof in question. Since the nth term is n(n+1)/2, the square number is 8[n(n+1)/2]+1 which equals to 4n^2+4n+1. We factorise this as follows:
4n^2+4n+1
= 4n^2 + 2n + 2n + 1
= 2n(2n+1) + 1(2n+1)
=(2n+1)(2n+1)
=(2n+1)^2
Since we have got (2n+1)^2 this means that for any value n which is a triangle number, 8n+1 equals to a square number.
However, it is not necessary that if 8n+1 is a square number, n has to be a triangular number.
In fact all the even numbers squared (example: 2^2, 4^2, 6^2 etc) contradict this statement.
If 8n+1 = 4
8n = 3
n = 3/8
If 8n+1 = 16
8n = 15
n = 15/8
If 8n+1 = 36
8n = 35
n = 35/8
Algebraically speaking:
8(n(n+1)/2) + 1 = (2n)^2 (Since 2n will always be an even number)
4(n^2 + n) + 1 = 4n^2
4n^2 + 4n + 1 = 4n^2
4n + 1 = 0
Therefore n = -1/4
Since n is negative AND not an integer, it is proven that n can not be a triangle number if 8n+1 is an even square number. We know that all triangular numbers will be integers because the sequence is n(n+1)/2 which means that the product of 2 consecutive integers is being divided by 2, and that means that one or the other will have to be even, thus being divisible by 2.
So not all the n values are a triangle number if the square number is equal to 8n+1.
But n is one if 8n+1 is an odd square number (example: 3^2, 5^2, 7^2 etc) .
If 8n+1 = 9
8n = 8
n = 1
If 8n+1 = 25
8n = 24
n = 3
If 8n+1 = 49
8n = 48
n =6
In an algebraic sense:
8(n(n+1)/2) + 1 = (2n+1)^2 (Since 2n is always even, 2n+1 will always be odd)
4n^2 + 4n + 1 = (2n+1)(2n+1)
4n^2 + 4n + 1 = 4n^2 + 2n + 2n + 1
Therefore,
4n^2 + 4n + 1 =4n^2 + 4n + 1
Since the left hand side of the equation is equal to the right hand side of the equation, it is proven that n will always be a triangle number if 8n+1 is an odd number.
Therefore, if 8n+1 is an ODD square number, n will be a triangular number. Not all square numbers, but only the odd ones.
Now, for the last question, to check if the given numbers are triangular numbers of not.
To be a triangular number, the numbers will have to give at least one integer answer when equated to the general form of a triangular number, n(n+1)/2. I say "at least one" because the answers will be quadratic so we will get 2 answers, and "integer" because - as stated before - all the numbers are going to be integer since either n or (n+1) will be even and thus divisible by 2.
6214:
n(n+1)/2 = 6214
n^2 + n = 6214*2
n^2 + n = 12428
n^2 + n - 12428 = 0
n = (-1 +- sqrt((-1)^2 - 4(1)(-12428)))/2
n = 110.98 , -111.98
Therefore 6214 is not a triangle number.
3655:
n(n+1)/2 = 3655
n^2 + n = 7310
n^2 - n - 7310 = 0
n = (-1 +- sqrt((-1)^2 - 4(1)(-7310)))/2
n = 85, -86
Therefore 3655 is a triangle number and is the 85th term in the sequence.
7626:
n(n+1)/2 = 7626
n^2 + n = 15252
n^2 - n - 15252 = 0
n = (-1 +- sqrt((-1)^2 - 4(1)(-15252)))/2
n = 123, -124
Therefore 7626 is a triangle number and is the 123rd term in the sequence.
8656:
n(n+1)/2 = 8656
n^2 + n = 17312
n^2 - n - 17312 = 0
n = (-1 +- sqrt((-1)^2 - 4(1)(-17312)))/2
n = 131.08, -132.08
Therefore 8656 is a not triangle number.