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The nonnegative integer solutions for the following equations:

$ x^3+1=y^2 $

x y
0 1
2 3

$ \frac{x(x+1)}{2}=\frac{y(y+1)(y+2)}{6} $

x y
0 0
1 1
4 3
13 8
47 18
9E 2X

$ x^2=\frac{y(y+1)(y+2)}{6} $

x y
0 0
1 1
2 2
E8 40

$ \frac{x(x+1)}{2}=\frac{y(y+1)(2y+1)}{6} $

x y
0 0
1 1
X 5
11 6
459 71

$ x^2=\frac{y(y+1)(2y+1)}{6} $

x y
0 0
1 1
5X 20

$ 2^x-7=y^2 $

x y
3 1
4 3
5 5
7 E
13 131

$ x!+1=y^2 $

x y
4 5
5 E
7 5E

$ x\#=y(y+1) $

x y
2 1
3 2
5 5
7 12
15 4E6

$ x^2-ny^2=1 $

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