Firstly, to prove that if T is a triangular number, 8T +1 is a square, I found the nth term of the triangular number sequence. This would give me a standard term to use and would be applicable to all triangular numbers.
Triangular number sequence: 1 3 6 10…
I noticed that this was a quadratic sequence, with the first differences being 1,2,3,4.. and the second common difference being 1.
1 3 6 10 a+b+c
2 3 4 3a+b
1 1 2a
I then used the above equations to solve and obtain the quadratic nth term:
2a=1 a=0.5
3(0.5) +b = 2
1.5 +b = 2
b = 0.5
a+b+c = 1
0.5+0.5+c = 1
c = 0
Therefore, the final nth term of the triangular number sequence is 0.5k^2 + 0.5k ( k being the term).
We need to prove that any triangular number *8+1 is a square number. Therefore,
8(0.5k^2 + 0.5k) +1 = 4k^2 + 4k +1
This can be factorised to (2k+1) (2k+1) = (2k+1)^2. Therefore, the conjecture is proved as the number (2k+1) is squared.
Secondly, to prove that 8k+1, as a square number, that n is a triangular number, I followed the steps below:
We have established that 8k+1 = 4k^2 + 4k +1
2) I then simplified this by subtracting 1 from both sides:
8k = 4k^2 +4k
Divide by 8:
k = 0.5k^2 + 0.5k
This is equivalent to the nth term of the triangular number sequence which is a general formula for all terms in the sequence. Therefore, with 8k+1 being square, k must be a triangular number.
Thirdly, to check that, for example, 6214, was a triangular number, I substituted it into 8k+1. If 8(6214) +1 was a square number, then 6214 (k) would be a triangular number. This is because we proved that a triangular number, when multiplied by 8 and add 1, would result in a square number.
8(6214) +1 = 49713
This cannot be a square number as no square numbers end in 3. Therefore, as 8k+1 is not square, k is not triangular - as 49713 is not a square number, 6214 is not a triangular number.
I repeated this for the following numbers:
b) 3655
8(3655) +1 = 29241
This is the square of 171, and therefore, 3655 is a triangular number.
c) 7626
8(7626) +1 = 61009
This is the square of 247, and therefore, 7626 is a triangular number.
d) 8656
8(8656) +1 = 69249
This has no integer solution, as the square root of 69249 is 263.152…
Therefore, as 69249 is not a perfect square, 8656 is not a triangular number.