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The Pythagorean Theorem is one of the most famous quadratic diophantine equations, which are equations with squared variables solved over the integers. How many solutions can you find?

Quadratic Diophantine Equations: Level 4 Challenges

$$4$$ has the property that if one adds it to double its square, it yields a perfect square. In other words for natural numbers $$m,n$$:

$n^2 + n^2 + n = m^2$

There are four such $$n<10^6$$. If $$4$$ is the smallest $$n$$, what is the second smallest $$n$$?

What is the largest integer $$\displaystyle n$$ for which $$n^2+24n+16$$ is a perfect square?

$$365$$ can be written as a sum of $$2$$ consecutive perfect squares and also $$3$$ consecutive non-zero perfect squares: $365=14^2+13^2=12^2+11^2+10^2$

What is the next number with this property?

For how many positive integers $$n<10^6$$ is $$2\times n! \times (n+2)!$$ a perfect square?

Find the sum of all positive integers $$m$$ such that $$2^m$$ can be expressed as sums of four factorials (of positive integers).

Details and assumptions

The number $$n!$$, read as n factorial, is equal to the product of all positive integers less than or equal to $$n$$. For example, $$7! = 7 \times 6 \times 5 \times 4 \times 3 \times 2 \times 1$$.

The factorials do not have to be distinct. For example, $$2^4=16$$ counts, because it equals $$3!+3!+2!+2!$$.

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