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While there have been many numbers that have been deemed 'lucky' or 'unlucky', 7 and 13 are two of the most prominently known.

So, this had led me to wonder if there were any connections between 7 and 13, and found the following.

$1$. $(7,13)$ is the largest integer pair $(n,m)$ for which ($n$th Fibonacci Number)=($n$th odd number)=$m$.

$2.$In the decimal expansion of $\log(7)$, which is $1.9459101490553\dots$, the $7$th digit after the decimal point is $1$, while the $13$th digit is $3$, giving us $13$.

Are there any other unique, special connections between $7$ and $13$?

By unique I mean that you cannot say something like $7$ and $13$ are both prime numbers or something of the sort.

Any other connections would be greatly appreciated if provided.

Community
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S.C.B.
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  • Interesting as it may sound, if one starts looking for such and other patterns, one can find many many examples – Shailesh Feb 16 '16 at 09:55
  • "By unique I mean that you cannot say something like..." is a very non-mathematical way of defining the term unique. You should state what this term stands for, not what it doesn't stand for. Assuming that you don't really have such definition at hand, your question can hardly be considered mathematical. – barak manos Feb 16 '16 at 09:59
  • Be careful, you're going down a dangerous path: https://www.youtube.com/watch?v=4ElUBr6-GLU – mathematician Feb 16 '16 at 10:14
  • The connection is that they are two prime numbers. Other connections "just happen". – Enrico M. Feb 16 '16 at 11:58
  • @barakmanos Sorry for being non-mathematical, I was just curious. I decided to close the question. – S.C.B. Feb 16 '16 at 13:27

1 Answers1

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You may found as many as you want. A little list

(1) There are $\mathbf{7}$ nondegenerate triangles you can make with sides in $\{1,2,3\}$ and $\mathbf{13}$ you can make with sides in $\{1,2,3,4\}$.

(2) Between $2^5$ and $2^6$ there are $\mathbf{7}$ primes, and in the next such interval from $2^6$ to $2^7$ there are $\mathbf{13}$ primes.

(3) They are the two smallest numbers which are not the difference between two primes.

(4) The two smallest primes $p$ such that the next prime is $p+4$.

(5) The two smallest primes $p$ such that $2p+1$ is composite.

(6) There are $\mathbf{7}$ bipartite graphs with 4 vertices and $\mathbf{13}$ bipartite graphs with 5 vertices.

(7) They are the two smallest $n$ such that $(\mathbf{7}^n+8^n)/(\mathbf{7}+8)$ is prime. Also true for $(\mathbf{13}^n+14^n)/(\mathbf{13}+14)$.

(8) The two smallest numbers $n$ such that $(n^2 + 1)/10$ is prime.

(9) The two smallest integers $n$ such that $\sin(n)<\cos(n)<\sin(n+1)$ (in radiants).

(10) The two smallest numbers $n$ such that the decimal part of $\sqrt{n}$ starts with digit $6=\mathbf{13}-\mathbf{7}$.

(11) The two smallest primes $p$ such that neither $x^3\equiv 2\pmod p$ nor $x^3\equiv 3\pmod p$ has a solution.

(12) The two smallest numbers $n$ such that $n^2$ (here 49 and 169) is a concatenation of two nonzero squares.

(13) The two smallest numbers such that $3^n$ ends with the same digit as $n$. Here $3^7=2187$ and $3^{13}=1594323$.

(13') $7=\lfloor e^2\rfloor$, and $13=\lfloor(e+1)^2\rfloor$.

(13") The sum of digits of $7^7\cdot 13^{13}$ and the sum of digits of $7^{13}\cdot 13^7$ are both equal to $7\cdot 13$. Probably they are the only two distinct numbers with this property.

Giovanni Resta
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