Five problems: a perfect square trinomial to rule them all

PROBLEMS

1. Show that there are infinitely many square triangular numbers.

2. Call an integer square-full if each of its prime factors occurs to the second power (at least). Prove that there are infinitely many pairs of consecutive square-fulls.

3. The prime decomposition of different integers $m$ and $n$ involve the same primes. The integers $m+1$ and $n+1$ also have this property. Is the number of such pairs $(m,n)$ finite or infinite?

4. Encuentre el número entero positivo $A$ que satisface $$A^{2} = 4\cdot 2006 + 4\cdot 2004 + 4 \cdot 2002 + \cdots + 4\cdot 4 + 4 \cdot 2 + 1.$$

5. Un entero positivo es bueno si es divisible entre todos sus factores primos al cuadrado. Por ejemplo, el $72$ es bueno ya que sus factores primos son $2$ y $3$ y $72$ es múltiplo de $2^{2}$ y $3^{2}$. Demuestre que hay una infinidad de parejas de números enteros consecutivos buenos.

REMARKS

I encountered these problems in different sources, on different ocassions. Clearly enough, they are not totally unrelated: for instance, the fifth problem is but the translation into Spanish of the second one and the sort of thing that we are asked to establish in all of them, with the exception of the fourth one, is more or less the same. Isn't it fabulous that we can establish each one of them by resorting to the fact that $4n(n+1)+1=(2n+1)^{2}$? (For example, a (beautiful) solution to the first problem goes as follows: if $n$ is a positive integer such that $\frac{n(n+1)}{2}$ is a perfect square, then $\frac{4n(n+1)[4n(n+1)+1]}{2}= 4\left(\frac{n(n+1)}{2}\right)(2n+1)^{2}$ is both a triangular number and a perfect square; since $1$ is a square triangular number, we are done.) Naturally, this is not to say that it is not desirable or possible to solve them in some other ways: in point of fact, both the first and the second problem can also be settled by considering a suitable Pell equation and the fourth one by determining the prime decomposition of $4(1003)(1004)+1$ (manually or with the aid of WA). The point here is that all of these problems can basically be solved by appealing to the squareness of $4n^{2}+4n+1$... Does any of you know of some other nice problem that can be disposed of by means of this trinomial?

ORIGINS

Problem 1: J. L. Pietenpol, A. V. Sylwester, Erwin Just, and R. M. Warten, Amer. Math. Monthly, 69 (Feb. 1962), pp. 168-169.

Problem 2: Donald J. Newman, A problem seminar. Springer Verlag, New York, 1982, p. 8.

Problem 3: You failed your math test, comrade Einstein. M. Shifman (editor). World Scientific, 2005, p. 24.

Problem 4: Problema 4 de la etapa semifinal estatal de la Vigésima Primera Olimpiada Mexicana de Matemáticas, (2007).

Problem 5: Tzaloa: Revista de la Olimpiada Mexicana de Matemáticas, (2013), no. 2, p. 26.

P.s. (April 9th, 2021) I have just noticed that I forgot to include the following problem in the above list: Prove that the equation $x^{2}+y^{2}+1=z^{2}$ has infinitely many integer solutions. Unfortunately, I don't recall at the moment where it was that I picked this one up from...