Numbers that are both square and triangular numbers were introduced in Chapter 1, and you...

Numbers that are both square and triangular numbers were introduced in Chapter 1, and you studied them in Exercise

(a) Show that every square–triangular number can be described using the solutions in positive integers to the equation x2 − 2y2 = 1. [Hint. Rearrange the equation m2 =.

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(b) The curve x2 − 2y2 = 1 includes the point (1, 0). Let L be the line through (1, 0) having slope m. Find the other point where L intersects the curve. .

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(c) Suppose that you take m to equal m = v/u, where (u, v) is a solution to u2 −2v2 = 1. Show that the other point that you found in (b) has integer coordinates. Further, .

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changing the signs of the coordinates if necessary, show that you get a solution to x2 − 2y2 = 1 in positive integers.

(d) Starting with the solution (3, 2) to x2 − 2y2 = 1, apply (b) and (c) repeatedly to find several more solutions to x2 − 2y2 = 1. Then use those solutions to find additional examples of square–triangular numbers. .

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(e) Prove that this procedure leads to infinitely many different square-triangular numbers. .

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(f) Prove that every square–triangular number can be constructed in this way. (This part is very difficult. Don’t worry if you can’t solve it.) 

Exercise: 

The first two numbers that are both squares and triangles are 1 and 36. Find the next one and, if possible, the one after that. Can you figure out an efficient way to find triangular–square numbers? Do you think that there are infinitely many?