Recently I have seen in an math forum this:
and just a few days later this one, too:
Pretty little tables, aren’t they? How could they be so regular? Can they be generalized somehow?
Answers are: yes, you will see, and yes.
Wonder #1
Let’s first take a look at the first table:
In order to understand it one has to work not with the specific numbers (digits), but with their abstract representations. For this, we will work with a number system of base (), which in the orginal tables is . Then we can rewrite several first members of the Table 1 as follows:
Ok, we see some regularity here. To proceed further, rewrite the row of that table in the form a mathematical equation , transforming the first beautifully looking number () into second beautifully looking number .
Here the series is:
Let’s go back to the main line of discussion.
Now we are interested in the following derivative series . The straightforward manipulation leads to the anticipated result:
i.e. . The initial pyramid of simple results holds for every base .
Example: for we have , so , , etc. Then, for example .
Wonder #2
Let’s look at the Table 2:
Here the first (i.e. the independent) variable is the exactly same as the one used for Table 1. The second (i.e. the dependent) variable is new one, determined by defining equation:
,
Then, using steps similar to these used in analysis of the Table 1, we get:
where there are copies of digit 1.
Nice. Easy.
Wonder #3
That series of relations has form where the dependent variable is
in the normal decimal system ().
For general basis this generalizes to the:
. Here the independent variable is
Now
For that covers all examples in the Table 3.
Note: Even more, we can add two more members to it, corresponding to and :
 corresponds to ,
 corresponds to .
Also, the provides one more line, which is prepended to this more complete table 3:
Table 3* 
 ,
 ,
 ,
 ,
 ,
 ,
 ,
 ,
 ,
 ,
 .

Wonder #4
Here the independent variable is
.
The resulting variable is
.
Now the sum in brackets can be transformed as follows:
which has three posible simplifications:
So, now we can write the final form for as following:
That is it.
Note that one can get some regularities for degrees higher than 2. For example, for degree 3 one has:
up to the last member (of that series) where the two central digits are the highest singledigit number allowed in the number system of base (i.e. if is odd, and if is even).
Note: Here , , , , , , , , , etc.
For the degree 4 the similar pyramid/table is:
Here one can also work some more (and I did not do that work yet) to establish which is the last member of that table (as a function of the base ), and what is the innermost digit in that last member.
This could be a homework for you. 🙂
For the degree 5:
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https://eikonal.wordpress.com/2012/03/14/moresimplemathwonders/  Mental calculation of cube root of a sixdigit number –
https://eikonal.wordpress.com/2010/01/14/mentalcalculationofcuberootofatwodigitnumber/  Squares with just two different decimal digits –
https://eikonal.wordpress.com/2010/01/05/squareswithjusttwodifferentdecimaldigits/  Number theory finite concidental sums –
https://eikonal.wordpress.com/2010/01/05/numbertheoryfiniteconsidentalsums/2011.07.06
Definitions:
 for every prime number
 (Leibnitz rule) for every two natural numbers
Concequences:
 .
 for any natural number one has .
 Eg: .
 in general
 for any prime number is equivalent of the exponential function’s property that its derivative is itself.
 If for prime and natural , then , , and .
 For infinitely many natural numbers there exist suitable s/t
 Ufnarovski and Åhlander give following conjecture: for every natural , as we observe its derivatives (as grows to infinity), the limit will be either , , or itself (if for some prime ).
 If , then .
 If , then (for all possible primes).
Sources:
 “Deriving the Structure of Numbers” by Ivars Peterson (Ivars Peterson’s MathTrek) – http://www.maa.org/mathland/mathtrek_03_22_04.html
 “How to differentiate a number” by Ufnarovski, V., and B. Åhlander (Journal of Integer Sequences 6; 2003.09.17) – http://www.cs.uwaterloo.ca/journals/JIS/VOL6/Ufnarovski/ufnarovski.html
Abstract: We define the derivative of an integer to be the map sending every prime to 1 and satisfying the Leibnitz rule. The aim of the article is to consider the basic properties of this map and to show how to generalize the notion to the case of rational and arbitrary real numbers. We make some conjectures and find some connections with Goldbach’s Conjecture and the Twin Prime Conjecture. Finally, we solve the easiest associated differential equations and calculate the generating function.
 “Investigations of the number derivative” by Linda Westrick – http://web.mit.edu/lwest/www/intmain.pdf
2010.12.10
A simple metric on the space of the publishing mathematicians. That space consists of the discrete nodes that represent each publishing mathematician. For each jointly publication there is a link between the two nodes representing authors of that publication. If two authors have N common publications, the link connecting them has weight N. A distance between two nodes (x and y) is defined as the shortest number of steps from x to y. The Erdos Number is the distance (metric) of x to the one specific node (Paul Erdos). Majority of the nodes in the space belong to a one extremely large cluster/graph (a dominant/principal component). Still, there are many solitary nodes.
2010.11.15
Definition: (source: [3]) A number with n digits, which, when multiplied by 1, 2, 3, …, n produces the same digits in a different order. For example, 142857 is a cyclic number: 142857 × 2 = 285714; 142857 × 3 = 428571; 142857 × 4 = 571428; 142857 × 5 = 714285; 142857 × 6 = 857142, and so on. It has been conjectured, but not yet proven, that an infinite number of cyclic numbers exist.
Properties:
 from [2]: … Multiplication by an integer greater than 7: adding the lowest six digits (ones through hundred thousands) to the remaining digits and repeat this process until you have only the six digits left, it will result in a cyclic permutation of 142857. Example:
Sources:
2010.11.02
Partition function
Definition: The partition function is defined by
.
Examples: The partitions of 5:
i.e. .
Properties:
 , , , , , , , , , , … , … , … , …
 An asymptotic formula: .
 Generating function: (where ).

 … [the] main contributions [to this integral] come from poles of , [and] these lie at every root of .
 Circle Method: using the function that estimates the contribution [to the above integral] from the poles on the circle ${\cal C}$ that are near the qth roots of unity. The main formula of the method is: . Taking the finite number of terms in that sum gives surprisingly good results, due to the fact that the estimated function has integer values.
Some applications of Circle Method:
 Lagrange’s conjecture: Each natural number can be expressed as the sum of at most four squares.
 Waring’s problem: Every sufficiently large natural number can be expressed as the sum of at most kth powers.
 Example: To calculate the exact expression for the number of ways () to write number as a sum of squares (i.e. ) use the th potency of the function :
 ,
 and, therefore:
Goldbach’s conjecture:
 Every even number greater than two can be represented as a sum of two primes.
 Every odd number greater than five can be represented as a sum of three primes.
Congruences for
Ramanujan’s congruences:
Congruences found in 1960s:
for .
Example: .
Theorem (Ono, 2000): For prime there exist infinitely many congruences of the form .
Theorem (Ahlgren, 2000): For prime and a positive integer there exist infinitely many congruences of the form .
Modular forms
Definition: Ramanujan’s taufunction:
Definition: for (with ).
The has modular symmetries: () for any integer that is unimodular (). In another words, is a modular form of weight 12.
Definition: The function is a modular form of weight if it satisfies the modularity equation (, ()).
The is a prototype of all modular functions. Some other examples are:
 ,
 .
Some properties of :
 if .
 Ramanujan’s conjecture: for every prime .
 Generalized Ramanujan’s conjecture: if has weight , then for all primes .
Congruences for function:
 for every prime .
 for .
 for .
 for .
 for .
 These are the only congruences of the form .
Sources
More
2010.01.05
There exist only finitely many squares with just two decimal digits” providing a really short list
plus infinite classes:
This is not correct. Thanks to Bruno Curfs for pointing this out, as well as for finding the reference [2] cited below.
 Namely, one can see that squares of all single digit numbers satisfy targeted condition:
 Then, there are squares of following twodigit numbers:
 ,
 ,
 ,
 ,
 ,
 ,
 ,
 ,
 ,
 ,
 ,
 For squares of 3digit numbers one finds:
 ,
 (thanks to Bruno Curfs)
Of the higher cases, there are known only infinitely long classes:
and following two “sporadic” occurrences:
 .
Author of page “1. Squares of 2 different digits” (http://www.asahinet.or.jp/~KC2HMSM/mathland/math02/three01.htm) provides an implementation of the algorithm to perform exhaustive check. According to his numerical experiments, there are no other sporadic occurrences up to at least (i.e. ).
Reference:
Similar here: More simple math wonders –
https://eikonal.wordpress.com/2012/03/14/moresimplemathwonders/  Mental calculation of cube root of a sixdigit number –
https://eikonal.wordpress.com/2010/01/14/mentalcalculationofcuberootofatwodigitnumber/  Squares with just two different decimal digits –
https://eikonal.wordpress.com/2010/01/05/squareswithjusttwodifferentdecimaldigits/  Number theory finite concidental sums –
https://eikonal.wordpress.com/2010/01/05/numbertheoryfiniteconsidentalsums/