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I will shortly be engaging with my 50th (!) birthday.

50 = 1+49 = 25+25 can perhaps be described as a "sub-Ramanujan" number.

I'm trying to put together a quiz including some mathematical content. Contributions most welcome. What does 50 mean to you?

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    Since your age is going from $49$ to $50$, I have posted as my answer below a proof that 49<50. We must therefore conclude that you're not getting younger.2014-07-14

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Let's see if I can prove that $49<50$.

The tangent line to the circle $x^2+y^2=1$ at the point where $x=y$ intersects the $x$-axis at $\sqrt{2}$. Lines with the same slope and a larger $x$-intercept do not intersect the circle; those with a smaller intercept are secant lines to the circle. Let's use $7/5$ as an approximation to $\sqrt{2}$. The line with $x$-intercept $7/5$ and slope $-1$ is seen to intersect the circle twice: at $(4/5,3/5)$ and at $(3/5,4/5)$. Therefore $ \frac75<\sqrt{2}. $ Squaring both sides, we get $ \frac{49}{25}<2. $ Multiplying both sides by $25$, we get $ 49<25\cdot2. $ So $49<50$.

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$50$ is the sum of three consecutive squares: $50=3^2+4^2+5^2$

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    I hadn't spotted this - which "ought" to be "obvious".2012-10-30
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$50$ is the least integer that is $1$ more than a square but is not squarefree.

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    Infinite. There are infinitely many integer solutions of $x^2+1=2 y^2$, for example.2012-10-31
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Fifty is the smallest number that is the sum of two non-zero square numbers in two distinct ways: $50 = 1^2 + 7^2 = 5^2 + 5^2$

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    @MarkBennet, incase you are pursuing this, i expect the two reasonable infinite families are the $pq$ I mentioned and $2 p^2.$ Note, though, that you get more examples by multiplying by any power of $4$ or of $r^2,$ with prime $r \equiv 3 \pmod 4,$ or mixing $4'$s and several different $r^2$'s.2012-10-30
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$50$ is the sum of the first $3$ Abundant numbers.

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    Made me learn about abundant numbers myself!2012-10-30
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There is an integral solution to the Mordell equation $y^2 = x^3 + 50$ with $x=-1,$ but nothing integral for $y^2 = x^3 - 50.$ There are rational solutions, however, beginning with $x=211/9.$ The group of rational points on this elliptic curve is infinite cyclic.

On the other hand, what 50 really means to me is that doctors in the U.S. begin to push you to do tests that are just, um, undignified.

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    I fear that indignity is an increasing part of the package!2012-10-30
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1 + 3 + 5 + 7 + 9 + 9 + 7 + 5 + 3 + 1 = 50

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    This is a consequence of $50=2\cdot 5^2$ and the well known identity $n^2=\sum (2k-1)$2012-11-04
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50 is half the sum of the first nine prime numbers

A lot more here: https://primes.utm.edu/curios/page.php?short=50