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This delightful animation by Stefan Nadelman depicts "the additive evolution of prime numbers", set to Lost Lander's song "Wonderful World": http://www.youtube.com/watch?v=TZkQ65WAa2Q. (If you haven't watched it, you may want to do so before reading the rest of the question.)

Frame from 2:05 in the video

Counting $1$ as prime, the video shows a prime number growing up to $23$ by absorbing smaller primes during the first verse and chorus. In the second verse, swarms of $23$s and $53$s feed to form all the remaining two-digit primes from $29$ through $89$. During the final chorus, these primes then join together to form the primes $97, 131, 331, 281, 251$, and finally $863$. The video demonstrates that $863$ is the sum of 15 consecutive primes:

$$863 = 29 + 31 + 37 + 41 + 43 + 47 + 53 + 59 + 61 + 67 + 71 + 73 + 79 + 83 + 89.$$

Is this just a coincidence? Is $863$ special? Or is there a reason to expect that a prime of about this size would be a consecutive sum of so many other primes?

Obviously there's a lot of room for artistic license in designing an animation like this one. I'm interested in the ways that non-trivial mathematical considerations constrain the artist to choose certain designs over others, because through those constraints, the art might teach us something about mathematics. I'd hate to think that it's just eye candy. So, in general, do you see any interesting patterns that the casual viewer might miss?

Edit: For reference, here are the sums represented in the video:

$$\begin{alignat*}{2} 1+1={}&2\\ 2+1={}&3\\ 3+2={}&5\\ 5+2={}&7\\ 7+1+1+1+1={}&11\\ 11+2={}&13\\ 13+1+1+1+1={}&17\\ 17+2={}&19\\ 19+1+1+1+1={}&23\\\\ 23+3+3={}&29\\ 23+3+5={}&31\\ 23+7+7={}&37\\ 23+5+13={}&41\\ 23+1+19={}&43\\ 23+7+17={}&47\\ 23+11+19={}&53\\\\ 53+1+5={}&59\\ 53+1+7={}&61\\ 53+1+13={}&67\\ 53+1+17={}&71\\ 53+1+19={}&73\\\\ 53+7+19={}&79\\ 53+7+23={}&83\\ 53+5+31={}&89\\\\ 29+31+37={}&97\\ 41+43+47={}&131\\\\ 59+61+67+71+73={}&331\\\\ 53+97+131={}&281\\ 79+83+89={}&251\\ 253+281+331={}&863\\ \end{alignat*}$$

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Just to add a bit here: $863$ is also the sum of $5$ consecutive primes $(863 = 163 + 167 + 173 + 179 + 181)$ and $7$ consecutive primes $(863 = 107 + 109 + 113 + 127 + 131 + 137 + 139)$. It's also a safe prime (meaning it takes the form $2p + 1$ where $p$ is prime) and a Chen prime (meaning $2p + 2$ is either prime or the product of two primes). –  yoknapatawpha Dec 21 '13 at 1:12
Did you discover the 15 terms sum equality by looking at the video?! –  Matemáticos Chibchas Dec 21 '13 at 3:18
@MatemáticosChibchas Sure! Not with mental math, though; I had to write down the number represented by each shape. Then, for example, if you recognize a $41$, a $43$, and a $47$, you know that they form a $131$ without having to study its anatomy. The fact that they're all primes made it easy to check my work and correct arithmetic errors. It was a little tedious, though. I'll edit the question to include the sums for reference. –  Chris Culter Dec 21 '13 at 3:32
Please don't consider $1$ as prime. –  Sammy Black Dec 21 '13 at 4:57
@SammyBlack Fear not, I don't. :) The artist does, presumably because "the additive evolution of non-composite numbers" doesn't sound as nice. –  Chris Culter Dec 21 '13 at 5:07

1 Answer 1

up vote 4 down vote accepted

The number of partitions increases exponentially, and, in this case, it's outright astronomical, so no, I don't think there's anything to it.

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Thanks for the suggestion! Do you have precise estimates, though? The number of partitions of $p_n$ increases very quickly with $n$, but the probability that a randomly chosen $k$-partition consists of consecutive primes should decrease very quickly with $k$. I'd be interested to know how these factors balance. –  Chris Culter Dec 21 '13 at 3:37
The partitioning wasn't chosen at random: It was cherry-picked out of $164,036365999,875151520$ unique representations. As far as sums of $15$ consecutive primes being themselves prime, there are $5$ such numbers in between $1$ and $1,000$, and $33$ lesser than $10,000$, etc. If we change the number of terms to $7$, we get a total of $17$ and $60$, respectively. If we make it $5$, we get $24$ and $116$. Even for a number of terms as high as $19$ we get $24$ lesser than $10,000$, starting at $857$. These are the facts. What you find so “fascinating” about them is beyond me... –  Lucian Dec 21 '13 at 4:18
Facts have patterns and, often, explanations. Why are there 5 up to 1000 and 33 up to 10,000? What is the density of primes near $n$ that can be written as the sum of $k$ consecutive primes? It's fine if we don't know, but it's more interesting if it's $2\gamma/n^{3/2}\log n$ as a consequence of Someone's Lemma. –  Chris Culter Dec 21 '13 at 5:05
Well... What is the sum of the first $15$ primes ? It can't be such a small number, can it ? :-) In fact, it's $328$. If we were to place further restrictions upon it, such as the sum being prime itself, then it becomes $379$. In the first $10^3$ numbers, there are $5$ such examples. So we'd expect about $50$ in the first $10^4$, but we get $33$, which makes sense, since the number of primes themselves increases logarithmically, not linearly. So we'd expect a maximum of $330$ up to $10^5$, and indeed only get $168$, for the same reason as above. Etc. –  Lucian Dec 21 '13 at 12:50
Thanks, that's a good start! I've posted a more specific question here. –  Chris Culter Dec 27 '13 at 6:43

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