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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$\begingroup$$50$ is the sum of three consecutive squares: $50=3^2+4^2+5^2$
$\endgroup$ 2 $\begingroup$$50$ is the least integer that is $1$ more than a square but is not squarefree.
$\endgroup$ 2 $\begingroup$Fifty is the smallest number that is the sum of two non-zero square numbers in two distinct ways: $$50 = 1^2 + 7^2 = 5^2 + 5^2$$
$\endgroup$ 6 $\begingroup$$50$ is the sum of the first $3$ Abundant numbers.
$\endgroup$ 2 $\begingroup$There is an integral solution to the Mordell equation $y^2 = x^3 + 50$ with $x=-1,$ but nothing integral for $y^2 = x^3 - 50.$ There are rational solutions, however, beginning with $x=211/9.$ The group of rational points on this elliptic curve is infinite cyclic.
On the other hand, what 50 really means to me is that doctors in the U.S. begin to push you to do tests that are just, um, undignified.
$\endgroup$ 1 $\begingroup$1 + 3 + 5 + 7 + 9 + 9 + 7 + 5 + 3 + 1 = 50
$\endgroup$ 1 $\begingroup$50 is half the sum of the first nine prime numbers
A lot more here:
$\endgroup$ $\begingroup$Let's see if I can prove that $49<50$.
The tangent line to the circle $x^2+y^2=1$ at the point where $x=y$ intersects the $x$-axis at $\sqrt{2}$. Lines with the same slope and a larger $x$-intercept do not intersect the circle; those with a smaller intercept are secant lines to the circle. Let's use $7/5$ as an approximation to $\sqrt{2}$. The line with $x$-intercept $7/5$ and slope $-1$ is seen to intersect the circle twice: at $(4/5,3/5)$ and at $(3/5,4/5)$. Therefore $$ \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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