## Sexy Primes: What Are They?

You are probably shocked about the title and you will probably think that this is a joke. Well, it’s not. There are really sexy primes and in fact, there are some research about them.

Sexy primes are prime numbers which differ by 6. The pair (5,11) are examples of sexy primes, while (7,13,19) are triplet sexy primes. Quadruplet and quintuplet primes also exist and examples of them are (5,11,17,23) and (5,11,17,23,29) respectively. Sadly, there is only one quintuplet sexy prime.

“No, I am not a sexy Prime.”

Just like the search for twin primes, the search for sexy primes are also in progress (especially for single mathematicians). As of this writing, the greatest sexy prime pair found has 11,593 digits (the first number) and the greatest prime triplet has 5132 digits (the first number).

In September 2010, Ken Davis found the largest quadruplet sexy prime yet, a 1004-digit with p = 23333 + 1582534968299.

The proof that there is one quintuplet prime is quite easy. It only involves finding testing all the possible remainders of integers when divided by 5 (recall equivalence classes) and prove that one of $(p, p + 6, p + 12, p + 18, p + 24)$ is prime.

Reference: Wikipedia, Image Credit: Optimus Convoy, Devian Art

## Domain and Range on a Graphical Perspective

Two weeks ago, I  discussed the basic concepts of domain and range which I presented in an ‘algebraic way.’ In this post, I would like to discuss these concepts from a graphical perspective.

The domain of a function $x$ is the set of points on the x-axis where if a vertical line is drawn, it will hit a point on the graph. Take for instance, in the linear function $f(x) = 2x$,  we are sure that we can always hit a point wherever we draw a vertical line. In algebraic explanation, we can always find an $f(x)$ for every $x$. Therefore, we can conclude the that domain of $f$ is the set of real numbers. On the other hand, if we draw a horizontal line and it hits the graph, then it is part of the range of the graph. Clearly, the range of the $f$ is also the set of real numbers.

## What is the shape in three dimensions?

Got this problem from a seminar yesterday, and I think it’s a pretty good exercise for visualization.  The bottom, side, and front views of an object is shown below.

How does the object look in three dimensions? There is a good discussion about this problem in StackExchange