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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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If you have the patience, you could find something here. –  Karolis Juodelė Oct 30 '12 at 21:17
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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? –  Steven Stadnicki Oct 30 '12 at 21:21
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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. –  MJD Oct 30 '12 at 21:42
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I agree with not closing; but I think this question is more suited to be CW if it stays open. –  Asaf Karagila Oct 30 '12 at 23:33
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There is currently a request to re-open. @MJD : you may want to add your vote too. –  Willie Wong Oct 31 '12 at 9:10

7 Answers 7

up vote 10 down vote accepted

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! –  Mark Bennet Oct 30 '12 at 21:20
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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. –  Will Jagy Oct 30 '12 at 21:30
    
@WillJagy I didn't have the next examples of squares in my memory bank, so huge thanks for that. –  Mark Bennet Oct 30 '12 at 21:32
    
@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. –  Will Jagy Oct 30 '12 at 21:38
    
@WillJagy That's a question for after the weekend. Mind you, I don't think I'll make it to cubes. –  Mark Bennet Oct 30 '12 at 21:40

$50$ 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 $3^2 + 4^2 = 5^2$ –  amWhy Oct 30 '12 at 21:24
    
I hadn't spotted this - which "ought" to be "obvious". –  Mark Bennet Oct 30 '12 at 22:20

$50$ 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? –  Willie Wong Oct 31 '12 at 9:08
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Infinite. There are infinitely many integer solutions of $x^2+1=2 y^2$, for example. –  Robert Israel Oct 31 '12 at 15:53

$50$ is the sum of the first $3$ Abundant numbers.

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Completely unexpected, and wonderful, answer. –  Mark Bennet Oct 30 '12 at 22:01
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Made me learn about abundant numbers myself! –  Jean-Sébastien Oct 30 '12 at 22:29

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! –  Mark Bennet Oct 30 '12 at 21:21

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 $n^2=\sum (2k-1)$ –  Daniel Littlewood Nov 4 '12 at 12:14

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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