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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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    Congratulations on turning $50$! Happy birthday.2012-10-30
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    If you have the patience, you could find something [here](http://oeis.org/search?q=50&language=english&go=Search).2012-10-30
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    While I congratulate you on your approaching birthday, this question seems awfully localized for this site. How many people are likely to be coming along looking for properties of the number 50?2012-10-30
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    @StevenStadnicki Well, if it is localised, close it. Unpacking the properties of particular integers can give some insight. I think the equations I gave also relate to $2x^2-y^2=\pm 1$ and the Pell Equation - well tested territory, but perhaps another perspective? The proposed answers may prove/disprove the "localised" hypothesis. I mentioned Ramanujan - wasn't there some quotation about particular numbers being "friends"? I put recreational maths as a tag - if there is no delight in it ... why bother?2012-10-30
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    I vote against closing this question. If you plan to vote to close, and my counter-vote has not yet been cancelled, please instead post a comment to the effect that you are cancelling this counter-vote.2012-10-30
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    I agree with not closing; but I think this question is more suited to be CW if it stays open.2012-10-30
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    [Ask Wolfram$\Alpha$!](http://www.wolframalpha.com/input/?i=50)2012-10-31
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    [There is currently a request to re-open](http://meta.math.stackexchange.com/a/6471/1543). @MJD : you may want to add your vote too.2012-10-31
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    It seems pretty clear at this point that the post has not solicited "debate, arguments, polling, or extended discussion", and instead that the answers are plainly factual. I think Qiaochu was wrong to close it prematurely.2012-10-31
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    I think it is right to make it CW, and I should have done that straight off - sorry. I don't mind it being closed, as it has generated enough interesting ideas to be going on with. Thanks to everyone who has responded.2012-10-31
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    I am strongly in favor of closing this question as too localized.2012-11-15
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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

8 Answers 8

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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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    You beat me to my contribution! :-) Indeed, note also that $3^2 + 4^2 = 5^2$2012-10-30
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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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    huh, that's a curious fact. Out of curiosity, is the set indicated in your answer known to be finite/infinite?2012-10-31
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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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    Online Encyclopedia of Integer sequences gives 4, 50, 1729, 635318657 (A016078). A much better reason to mark the occasion than some mere multiple of 10!2012-10-30
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    @MarkBennet , something is off with your list, which should include $65 = 1 + 64 = 49 + 16$ and $85 = 81 + 4 = 49 + 36,$ in fact any product $pq$ with primes $p,q \equiv 1 \pmod 4, \; \; p \neq q.$ Also $1729 = 7 \cdot 13 \cdot 19 $ is not the sum of two squares. Two cubes, yes, in two different ways.2012-10-30
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    @WillJagy I didn't have the next examples of squares in my memory bank, so huge thanks for that.2012-10-30
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    @MarkBennet, I see what they did, "Smallest number that is sum of 2 positive n-th powers in 2 different ways." Evidently they do not know if this is possible for exponent 5.2012-10-30
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    @WillJagy That's a question for after the weekend. Mind you, I don't think I'll make it to cubes.2012-10-30
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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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    Completely unexpected, and wonderful, answer.2012-10-30
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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