## Integermania!

#### Curiosities

The webmaster of this site has received a few interesting submissions over the year which are worth sharing, but won't always qualify on the other pages due to their exquisiteness level. Therefore, to qualify for inclusion on this page, a submission must tickle the webmaster's fancy.

 $\dfrac{\sigma(\Gamma(5) + 7 - 3)}{2} = 28$ A perfect solution. A perfect number is defined as a number which is equal to the sum of its proper divisors. If all divisors are included, the sum will be twice the number. This level 6.4 First Four Primes solution incorporates the definition of a perfect number for the second perfect number, 28. Submitted by Chris Imm of Prairie Village, Kansas, in December, 2001. $10^{(3/.3)^{\Gamma(3)}} = \text{googol}$ Googol and still counting. This level 5.4 Letters of JCCC solution was the first of several googol solutions submitted by Brian Hollenbeck of Emporia, Kansas, in March, 2002. $\dfrac{\arccos(.5)}{(2 \times 3)^{\circ}} + 7 = 17$ Non-trivial trigonometry. This level 4.6 First Four Primes solution uses an inverse trig function to produce a 60-degree angle. Submitted by Mikel King of Paola, Kansas, in May 2002. $\tan \left(\left( \Gamma\left( \dfrac{3}{7-5} \right) \right)^2 \right) = 1$ Gamma at a half-integer. Since the gamma function and the factorial function are related by a horizontal translation (graphically speaking), the gamma function forms a very convenient alternative to the factorial for Integermania. But the factorial is defined only for non-negative integers, whereas the gamma function is defined on the entire complex plane, except for the non-positive integers. This level 5.4 First Four Primes solution uses the value of the gamma function at a half-integer. Submitted by Andrew Bennett of Manhattan, Kansas, in May, 2003. $\pi(d(((p(p(0!)))!)!)) = 10$ Creation out of nothing. With enough unary functions, all of the integers could probably be created from the singleton set {0}. Number theory appears to provide enough variety to do it using at most level 6 functions (although the unary surcharge eventually creates higher level solutions). This level 7.4 solution from the set {0} suggests how it could be done. In this problem, $p(n)$ is the $n$th prime, $d(n)$ is the number of divisors of $n$, and $\pi(n)$ is the number of primes at most $n$. Submitted by the webmaster, June, 2003. $- \log \sqrt{0! \% \cdots \%} = n$ Creation out of nothing, completed. With $n$ percent signs, this expression equals $n$. Thus all the integers have been created, albeit at level $4.8+0.2n$. Submitted by the webmaster, May, 2012.