2015 the year , the number , and some of their characteristics

The year 2015 has begun (wish it will be a happy year for everybody) and here are some properties of the corresponding number (worked out with the help of Mathematica).

First , a list of the representations of 2015  in various number base notations:

2015 base notation

2015 in roman numerals: MMXV;

2015 can be represented as:

2015=2^{11}-33

2015=\frac {92^4 - 1} {35553}

2015-relation-1 2015-relation-2

Here is a big one (obtained with Mathematica by solving x^3 + x^2 + x = 2015 ):

2015-relation-3

2015=5 x 13 x 31

2015-trig-1

The last relation above can be  derived from the trigonometric equation:

2015-trig-2

I have mentioned these types of trigonometric relations in a previous post.

The number of prime numbers up to 2015 is 305 , the last  prime number before 2015 is 2011 ,  and the prime number which comes after 2015 is 2017.

The last Mersenne prime number before 2015 is:
127=2^{7}-1

The Mersenne prime number which comes after 2015 is:
8191=2^{13}-1

The last Sophie Germain prime (n is a Sophie Germain prime if 2n + 1 is also prime) before 2015 is 2003 , and the Sophie Germain prime which comes after 2015 is 2039.

The Riemann prime counting function is given by:

Riemann prime counting function

where li(x) is the logarithmic integral and  μ(n) is the Möbius function.

And below is a graph showing the natural distribution and the number  of prime numbers π(n) less than or equal to n (blue colored curve , here n=2015) along with the distribution of primes given by the Riemann counting function (orange colored curve) up to 2015 :

 prime-riemannr

 

Moving to other fields of science , I will mention briefly a few events taking place in the year 2015:

In the field of space exploration , the New Horizons spacecraft will get closest to the dwarf planet Pluto in July 2015.

In the following image (made with the help of Starry Night and Photoshop) the longer red arrow near the name of a planet indicates the direction towards the place where the planet is located on its orbit.

Pluto and New Horizons

The Dawn spacecraft will arrive at the dwarf planet Ceres between Mars and Jupiter in March 2015.

Concerning astronomy and sky events , there will be a solar eclipse in March 20 , a lunar eclipse in April 4 , a partial solar eclipse in September 13 , and a lunar eclipse in September 28.

And that is it for now.

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