INTRODUCTION
Trying to learn about Riemann, his zeta function and hypothesis, I ended up reading about Dirichlet. Reading about him, I found in Wiki that the German mathematician Carl Friedrich Gauss (1777–1855) said,
“Mathematics is the queen of the sciences
and number theory is the queen of mathematics.”
I thought, of course, I, a retired math teacher, must learn about the queen of mathematics. So I decided to start this page with information about this queen.
I like “numberphiles” introduction but this page will take my first Number theory information from Bullis Student Tutors (see below).
The greatest collection online of Number theory material is found in
A good introduction of Number Theory has been done by Robin Wilson from Gresham college in this Youtube lecture:
INDEX
 introduction
 Math letters and number theory symbols (link to a separate page)
 Arithmetic
 The number system
 Number theory areas
 Elementary
 Infinite number series
 Euler´s zeta function
 Riemann hypothesis
 Pascal triangle
 The Ramanujan infinite series and summation
 R summation explained
 Bernoulli numbers
 Analytical
 Algebraic
 Geometric
 1729, the Ramanujan number
 Probabilistic
 Algorithmic
 Modular arithmetic
 Number theory applications
 Conclusion
 Sources
Aritmetic
Fundamental_theorem_of_arithmetic
also called the unique factorization theorem or the uniqueprimefactorization theorem states that every integer greater than 1^{} either is a prime number itself or can be represented as the product of prime numbers … For example,
The theorem says two things for this example: first, that 1200 can be represented as a product of primes, and second, that no matter how this is done, there will always be exactly four 2s, one 3, two 5s, and no other primes in the product. Read more in https://en.wikipedia.org/wiki/Fundamental_theorem_of_arithmetic
The number system
can be visualized with this set of complex numbers with its subsets
(Source: Wiki )
 imaginary number i
is a complex number that can be written modular that has a real number multiplied by the imaginary unit i,^{} which is defined by its property i^{2} = −1. Read more below.  complex numbers a + bi
A complex number is a number that can be expressed in the form a + bi, where a and b are real numbers, and i is a solution of the equation x^{2} = −1. Because no real number satisfies this equation, i is called an imaginary number. Despite the historical nomenclature “imaginary”, complex numbers are regarded in the mathematical sciences as just as “real” as the real numbers.
 real numbers ℝ
( include all the rational numbers, such as the integer −5 and the fraction 4/3, and all the irrational numbers) rational numbers set (ℚ)
that can be expressed as the quotient or fraction p/q of two integers,  Bernoulli numbers B_{n} are a sequence of rational numbers
 integers set ℤ^{+}
consists of zero (0), the positive natural numbers (1, 2, 3, …), also called whole numbers or counting numbers and their additive inverses (the negative integers, i.e., −1, −2, −3, …, when added to a yields zero natural number set ℕ^{+} (used for counting (3, 2,1, 0, 1, 2, …)
 positive
 negative
 prime numbers
“Those numbers you can’t divide into other numbers, except when you divide them by themselves or 1?”
 composite numbers
All composite numbers are made up of, and can be broken down (factorized) into a product (a x b) of prime numbers. Prime numbers are in this way the “building blocks” or “fundamental elements” of numbers.
( Jørgen Veisdal in medium.com/cantorsparadise ) odd
 even
 triangular numbers
if that number of pebbles can be arranged in a triangle (Joseph H. Silverman, “What is number theory” )  perfect numbers
 when the sum of all its divisors, other than itself, back up to the original number.
(like 1 + 2 + 3 = 6 and 1 + 2 + 4 + 7 + 14 = 28 and 496 and more)  (Joseph H. Silverman, “What is number theory” )
 square numbers
if that number of pebbles can be arranged in a square (ℤ^{2}, like 4, 9, 12,…)
 irrational numbers
Whose decimal expansion does not terminate, nor end with a repeating sequence. Examples are: √2 (1.41421356,
 π (3.14…..),
 Euler’s e (2,71828 18284 59045 23536….)
 the golden ratio φ (1,6180339887….)
 in fact all square roots of natural numbers, other than of perfect squares (ℝ^{2})
 Surreal numbers
Research on the go endgame by John Horton Conway led to the original definition and construction of the surreal numbers containing the: real numbers ℝ
 infinite numbers
larger or smaller in absolute value than any positive real number  infinitesimal numbers,
quantities that are closer to zero than any standard real number, but are not zero
respectively
 Transfinite numbers
In mathematics, transfinite numbers are numbers that are “infinite” in the sense that they are larger than all finite numbers, yet not necessarily absolutely infinite
(with the symbol: Ω. the extension of the idea of infinity proposed by mathematician Georg Cantor. Cantor linked the Absolute Infinite with God,) These include: the transfinite cardinals,
which are used to quantify the size of infinite sets,
which appeared as the subscripts α in S_{α}
 the transfinite ordinals,
which are used to provide an ordering of infinite sets. The term transfinite was coined by Georg Cantor in 1915
 the transfinite cardinals,
 rational numbers set (ℚ)
Ordinal numbers
Ordinals were introduced by Georg Cantor in 1883^{[1]} in order to accommodate infinite sequences and classify derived sets
.
Perfect numbers
Image is taken from Robin Wilson, Gresham college in this Youtube lecture. See in my introduction
..
Prime numbers
The sieve of Eratosthenes – How to find the prime numbers
This is a “an ancient algorithm for finding all prime numbers up to any given limit.”
Read more in https://en.wikipedia.org/wiki/Sieve_of_Eratosthenes
(60=2x2x3x5) or (60=5x3x2x2)
, and that this can be done in only one way apart from the order that the primes appear.
So we could have written 60 as two times five times three times two, or some other order,
but there must always be two twos, one three, and one five.
(60=2x3x1x5) or (60=2x3x1x1x5)
Proofs of the theorem were later made by:
 Euler
 Erdős
 Furstenberg
 de Polignac
 Dirichlet
 Bertrand–Chebyshev
 Another mathematician who worked with primes were:
 Marin Mersenne
Image is taken from Robin Wilson, Gresham college in this Youtube lecture. See in my introduction.
 Marin Mersenne
 Fermat
Image is taken from Robin Wilson, Gresham college in this Youtube lecture. See in my introduction.
Checking method. Is a number a prime number?
ZachStar shares a special method:
..
Imaginary numbers in quantum waves
What the hell are imaginary numbers? AsTibee describes it in the video below , I have trouble “seeing imaginary numbers.” I have understood that imaginary numbers however exist in real life.
I have found that a wave that does not change in amplitude like a quantum wave, can only be described with an imaginary number whose square is 1 ( i x i = 1 that is i^{2} = −1 )
Livescience ) says:
Imaginary numbers, also called complex numbers, are used in reallife applications, such as electricity.
… Imaginary numbers can also be applied to signal processing, which is useful in cellular technology and wireless technologies, as well as radar and even biology (brain waves). … Essentially, if what is being measured relies on a sine or cosine wave, the imaginary number is used. ( Livescience )
The wave has a complex amplitude.
The wave amplitude is actually a complex number
www.pitt.edu/~jdnorton shares a very good explanation of what imaginary numbers are and where these are useful. The article says among others:
“the amplitude of quantum wave is a complex number. .. It will turn out that it is very useful for us when we want to represent how waves propagate in space according quantum theory.” Compare these two differences:
 position for a particle: “add one for each unit of time.”
Later particle position 
=  distance moved (one for each unit of time) 
+  Initial particle position 
2. state for a wave: “multiply by i for each quarter period”
multiplying a wave by a factor is very close to what we need to move the wave forward in time. Where it goes wrong is that we are (tacitly) using real numbers as amplitudes.
Later wave state 
=  phase factor (scaled to i for each quarter wave period) 1 (start) 
x  Initial wave state 
“For this special case (ed. a quantum wave), the magnitude is constant. In the figure this is represented by the constant length of the amplitude arrow. The phase, however, advances uniformly with distance. That is, as we advance along the wave(ed. From left to right in the image below), the amplitude cycles through the values 1, i, 1, i, 1 and, of course, all the infinitely many values in between. What results is a wave that is represented by the helical (“corkscrew”) shape below:
.
Infinity
Infinity with the symbol ∞ is not a number but a concept. Like George Cantor explained, “There are different kinds of infinity”. Some are bigger and some are smaller.
Some can be listed
 1,2,3,4,5,6,…..
 1,1,1,1,1….
fractions and decimals can be listed
Real numbers can not be listed?
Numberphile has a good presentation about infinity
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Number theory introduction
Number theory is the study of the set of positive whole numbers 1, 2, 3, 4, 5, 6, 7, . . . , which are often called the set of natural numbers. We will especially want to study the relationships between different sorts of numbers. Since ancient times, people have separated the natural numbersℕ into a variety of different types. Here are some familiar and notsofamiliar examples:
 odd 1, 3, 5, 7, 9, 11, . . .
 even 2, 4, 6, 8, 10, . . .
 square 1, 4, 9, 16, 25, 36, . . .
 cube 1, 8, 27, 64, 125, . . .
 prime 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, . . .
 composite 4, 6, 8, 9, 10, 12, 14, 15, 16, . . .
 1 (modulo 4) 1, 5, 9, 13, 17, 21, 25, . . .
 3 (modulo 4) 3, 7, 11, 15, 19, 23, 27, . . .
 triangular 1, 3, 6, 10, 15, 21, . . .
 perfect 6, 28, 496, . . .
 Fibonacci 1, 1, 2, 3, 5, 8, 13, 21, . . .
(read more at www.math.brown.edu )
It has been difficult to find a good Number theory introduction in Youtube. The best one I have found is this:
Its says among others that
Number theory called Higher arithmetic (from the Greek ἀριθμός arithmos, ‘number’ and τική [τέχνη], tiké [téchne], ‘art’) is essentially a study of
 the mathematical interactions between the number types in the number system.
 the mathematical interactions of the imaginary numbers within the number system
Number theory asks questions about the relationships between the number types.
Euclide looked at prime numbers. He studied how many prime numbers exist. He came to a formal proof that was a proof by contradiction (saying that there is a finite number of primes and showed why this was wrong)
Number theory has a lot of realworld applications.
Fundamental Theorem of Arithmetic (FTA)
This says that all numbers can be factored to a unique set och prime numbers. E.g. 12 can be factorized with the primes 2 and 3 like
12 = 3x2x2
Number theory is used for cryptography and largescale networks.
Encryption uses unique prime factorization.
In July 2021 I discovered another introduction that is also worth listening to.
t is held by Robin Wilson from Gresham college in this Youtube lecture:
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Set theory
“A set is a Many which allow itself to be thought of as a One” — Georg Cantor
The idea of the set was shared by Georg Ferdinand Ludwig Philip Cantor with his Cantor set.
Set is a collection of objects called elements
A example of a set
The intersection of two sets A and B is the set of elements that are in both A and B. In symbols .
^{}For example, if A = {1, 3, 5, 7} and B = {1, 2, 4, 6} then A ∩ B = {1}.
The number 9 is not contained in the intersection of the set of prime numbers {2, 3, 5, 7, 11, …} and the set of even numbers {2, 4, 6, 8, 10, …}, because 9 is neither prime nor even.
A more elaborate example (involving infinite sets) is:
 A = {x is an even integer}
 B = {x is an integer divisible by 3}
 Set builder notation
here are two examples taken from https://www.youtube.com/watch?v=tyDKR4FG3Yw  1. A set of ℚ as { 1/2, 1/3, 1/4, 1/4 …} can be written as
 ℚ = { m/n  m,n ∈ ℤ  n ≠ 0 }
 2. A set of even integers as ℚ = { 4 , 2 , 0 , 2, 4, …}
can be written as 2ℤ= {2n  n ∈ ℤ} 
Cardinality of set
“the cardinality of a set is a measure of the “number of elements” of the set. For example, the set contains 3 elements e.g. {2, 4, 6}, and therefore has a cardinality of 3” ( Wiki ). This can be written as
 = 3
finite set
In mathematics, a finite set is a set that has a finite number of elements. Informally, a finite set is a set which one could in principle count and finish counting. For example,
The set of all integers, {…, 1, 0, 1, 2, …} is a countably infinite set… The set of all real numbers is an uncountably infinite set. The set of all irrational numbers is also an uncountably infinite set
“In mathematics, the Cantor set is a set of points lying on a single line segment that has a number of remarkable and deep properties. It was discovered in 1874 by Henry John Stephen Smith^{[1]}^{[2]}^{[3]}^{[4]} and introduced by German mathematician Georg Cantor in 1883. Through consideration of this set, Cantor and others helped lay the foundations of modern pointset topology. “

Russell’s paradox
In the foundations of mathematics, Russell’s paradox (also known as Russell’s antinomy), discovered by Bertrand Russell in 1901,^{[1]}^{[2]} showed that some attempted formalizations of the naïve set theory created by Georg Cantor led to a contradiction. ( Wiki )
ohn Horton Conway talkes about Cantor in this video:
In set theory, you find discussions also about functions like
 noninjective functions
 bijections like
 injective functions and
 surjective functions
 nonsurjective functions
In mathematics, an injective function (also known as an injection, or onetoone function) is a function that maps distinct elements of its domain to distinct elements of its codomain. In other words, every element of the function’s codomain is the image of at most one element of its domain. Look at this injective function and its x and y sets.
 Comments:
Y ∈ Y_{2 } (“Y is an element of Y2 “)  X_{1} ⊂ X (“X_{1 }is subset of X“)
Number theory areas
This is a translation from the Swedish presentation in Wikipedia, translated by me to English.
Traditionally, number theory is the branch of mathematics that deals with the properties of integers. Number theory has evolved to become an accepted technique for addressing problems even in other branches of mathematics.
Number theory can be divided into several areas depending on the methods used and the problems being investigated:
 .Elementary number theory
 Analytical number theory
 Algebraic number theory
 The geometry of number theory
 Probabilistic number theory
 Algorithmic number theory
.
1. Elementary NT
In elementary number theory, integers are studied without the use of any of the techniques from other areas of mathematics. To elementary number theory belongs questions like (links to Wiki):
 division algorithm.
 Euclidean’s algorithm for calculating the greatest common divisor,
 factorization of integers in prime numbers,
 examination of perfect numbers and congruences
Examples of the elementary number theorems are(links to Wiki):
 Fermat’s last theorem,
 Euler’s theorem,
 the Chinese reminder class
 and the quadratic reciprocity theorem.
included are also an investigation of the properties of arithmetic functions such as(links to Wiki):
 the Hungarian Möbius function
 and Euler’s totient (φ) function
 as well as integer sequences such as faculties and Fibonacci numbers.
Faculty in mathematics is a function. For an integer greater than zero, the faculty is equal to the product of all integers from 1 up to and including the number itself.
Faculty is expressed as expressed as
0  1 
1  1 
2  2 
3  6 
4  24 
Many questions in elementary number theory are exceptionally difficult and require entirely new approaches. A few examples are:
 Goldbach conjecture that every even integer greater than 2 can be expressed as the sum of two primes.
 Catalan’s Conjecture on 2^{3} and 3^{2} (8, 9)
 Alphonse de Polignac’s conjecture that states that there are infinitely many cases of two consecutive prime numbers with difference n.
 Collatz’s assumption of simple iteration
 diophantine equations that have proved “insoluble”. See also Hilbert’s tenth problem.
The outstanding problem left over from Gauss’ Disquisitiones Arithmeticae concerning the solutions to the indeterminate diophantine equation ax²+by²+cz² = 0, also known as Legendre’s equation. ( medium.com/cantorsparadise/thenatureofinfinityandbeyond )
)Michael Penn, Professor of Mathematics at Randolph College in Lynchburg, Virginia makes a good presentation of the division algoritm,
 The division algorithm states that
a=bq+r where ab∈
 q is the quotient and r the remainder
 0≤r<b a
 nd b>0 is valid. E.g:
 if b=2 then a=21=2×10+1
 if b=6 then a=35=6×5+5
 integers are denoted with a doublestruck boldface Z (Unicode 2124)
 x∈ is the same as to say that “x is an element of “
For the proof a set S, is to be considered.
S=⌈abxx∈abx∉≥0⌉
Michael Penn is proceeding very fast. One that does it slower is “learnable” in this video
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infinite number series
I think that the best introduction to infinite series is the one made by 3 Blue1Brown in this youtube:
Eulers zeta function
For s ⊂ ℝ It an be written as
 Read bout its derivation in https://empslocal.ex.ac.uk
 With s=1 it can be reduced with s⊂ℝ =1 to the harmonic series that can be “visualized” in music with overtones of a vibrating string:
 Source: www.youtube.com/watch?v=XPbLYD9KFAo

 Zeno’s paradox
Riemann hypothesis
Rieman developed Eulers zeta function to include complex numbers ( s=a + bi ).
The prime number series
2, 3, 5, 7, 11, 13, 17, 19, 23, 29, p. What is p? 31. What is the next p? It’s 37. The p after that? 41. And then? 43. How, but… …how do you know what comes next?
Present an argument or formula which (even barely) predicts what the next prime number will be (in any given sequence of numbers), and your name will be forever linked to one of the greatest achievements of the human mind, akin to Newton, Einstein, and Gödel. Figure out why the primes act as they do, and you will never have to do anything else, ever again…
Riemann’s hypothesis about the roots of the zeta function, however, remains a mystery.” (Jørgen Veisdal, medium.com )
This teacher tells about the Riemann hypothesis in a clear way but I found one that maybe is even better. See below just above the section of the Ramanujanserie section.
Bernoulli could calculate easily the sum of 1^{10}+2^{10}+3^{10}….+1000^{10}
(Source: Mathologer in www.youtube.com )
The deriving formula for number series
“Mathologer” makes a great introduction to Bernoulli number series and how to calculate these in this video https://www.youtube.com/watch?v=fw1kRz83Fj0
Bernoulli calculated this sum in a few minutes buy hand: How did he do? Noone in Mathologers video comments has supplied an answer
“Mathologer” shows graphically in a splendid way how he first approximates and then gets exact answers to several series.
IMPORTANT. All following illustrations are screenshots from Mathologer in www.youtube.com.
The series Mathologer illustrates are:
 1+2+3+4+5+6+7+8+9+10 = (10^{2}+10)/2
(Tip: the series forms a triangle)
⇒1+2+3+..+n =(n^{2}+n)/2 =n(n+1)/2  1+2+3+4
(This is a simpler series. To get an exact answer we use two of these triangles.)2(1+2+3+4) = The area will be 4(4+1)/2
so for 2(1+2+3+4+…+n) we get n(n+1)/2  1^{2}+2^{2}+3^{2}+4^{2
}This serie forms a Mayan square pyramid. The volume formula for this Pyramid is a third of a base area * height. V_{p}=A_{b}*h/3
With this formula, we get an approximate volume of 4^{3}/3. To get an exact volume we must use 6 pyramids to make a to build a parallelepiped.
The volume of six pyramids becomes
V_{p}=Ab*h= 6(1^{2}+2^{2}+3^{2}+..4^{2})=4(4+1)(2*4+1)
As the parallellepiped is made of 6 pyramids, the volume of 1 pyramids is therefore A_{b}*h = 4 *(4+1)* (2*4+1) / 6
For a pyramid build of the serie 1^{2}+2^{2}+3^{2}+4^{2}+..+n we get therefore the volume
4*(n+1)*(2n+1)/6
 1^{3}+2^{3}+3^{3}+..4^{3
}
1^{3}+2^{3}+3^{3}+..4^{3 }gives the formula (1+2+3+4)^{2 }For the serie (1^{3}+2^{3}+3^{3}+..4^{3}+..+n) we get the formula (1+2+3+..+n)^{2}
Observe that 1^{3}+2^{3}+3^{3}+4^{3} =(1+2+3+4)^{2}
As 2*(1+2+3+4)=4*(4+1)/2
So ⇒ 1^{3}+2^{3}+3^{3}+4^{3 }= (4*(4+1)/2)^{2}
For a infinite cubic serie we get
1^{3}+2^{3}+3^{3}+4^{3} +..+n^{3}= (n(n+1))/2)^{2}
Findiing patterns
(Source: Mathologer in www.youtube.com )
Expanding the formulas we get
1^{0}+2^{0}+3^{0}+4^{0 }+..+n^{0} 
n  S_{0}  
1^{2}+2^{1}+3^{1}+4^{1 }+..+n^{1}  (n(n+1))/2  S_{1}  (1/2)^{2}+(1/2)n 
1^{2}+2^{2}+3^{2}+4^{2 }+..+n^{2}  (n(n+1)) (2n+1))/6  S_{2} 
(1/3)^{3}+(1/2)n^{2} 
1^{3}+2^{3}+3^{3}+..4^{3}+..+n^{3}  ((n(n+1))/2)^{2}  S_{3}=(S_{1})^{2} 
1/4)^{4}+(1/2)^{3}+(1/4)n^{2} 
Following the pattern above we get
1^{4}+2^{4}+3^{4}+..4^{4}+..+n^{4}  1/5)^{5}+(1/2)^{4}+????? 
The sums can be expressed with the lower sums as with S_{3}=(S_{1})^{2}
To get the rest of the 4 dimension series formula, we use an algebraic proceeding.
Binomial identities
is an identity that connects different powers.
“Special cases of the binomial theorem were known since at least the 4th century BC when Greek mathematician Euclid mentioned the special case of the binomial theorem for exponent 2.^{[1]}^{[2]} There is evidence that the binomial theorem for cubes was known by the 6th century AD in India.” (Source: wiki )
.
Pascal triangle
In the binomial identity, the coefficients are taken from the Pascal triangle.
The binomial identity may be shown as (using the (x1) version:
 (x1)^{0} = 1x
 (x1)^{1} = 1x^{1}1
 (x1)^{2} = 1x^{2}2x^{1}+1
 (x1)^{3} = 1x^{3}3x^{2}+3x^{1}1
 (x1)^{4} = 1x^{4}4x^{3}+6x^{2}4x^{1}+1
 (x1)^{5} = 1x^{5}5x^{4}+10x^{3}10x^{2}+5x^{1}1
Moving all x^{5 }to the right we get:
+5x^{4}10x^{3}+10x^{2}5x^{1}+1 = x^{5}–(x1)^{5} Substituting X with 1,2,3,4 get these four equations: +5*1^{4}10*1^{3}+10*1^{2}5*1^{1}+1 = 1^{5}–(11)^{5 }+5*2^{4}10*2^{3}+10*2^{2}5*2^{1}+1 = 2^{5}(21)^{5 } +5*3^{4}10*3^{3}+10*3^{2}5*3^{1}+1 = 3^{5}(31)^{5} +5*4^{4}10*4^{3}+10*4^{2}5*4^{1}+1 = 4^{5}(41)^{5} Simplifying this we get: +5*1^{4}10*1^{3}+10*1^{2}5*1^{1}+1 = 1^{5}^{ }+5*2^{4}10*2^{3}+10*2^{2}5*2^{1}+1 = 2^{5}–1^{5 } +5*3^{4}10*3^{3}+10*3^{2}5*3^{1}+1 = 3^{5}2^{5} +5*4^{4}10*4^{3}+10*4^{2}5*4^{1}+1 = 4^{5}3^{5} Adding these 4 equations we get the sum where the marked in red an blue cancel out so only 4^{5}remains on the right side. +5*1^{4}10*1^{3}+10*1^{2}5*1^{1}+1 = 1^{5}^{ }+5*2^{4}10*2^{3}+10*2^{2}5*2^{1}+1 = 2^{5}–1^{5 } +5*3^{4}10*3^{3}+10*3^{2}5*3^{1}+1 = 3^{5}2^{5} +5*4^{4}10*4^{3}+10*4^{2}5*4^{1}+1 = 4^{5}3^{5 } ^{—}^{—}^{—}^{—}^{————}^{————}^{————}^{———— }We can write the sum as +5*1^{4}10*1^{3}+10*1^{2}5*1^{1}+1 = ^{ }+5*2^{4}10*2^{3}+10*2^{2}5*2^{1}+1 =^{ } +5*3^{4}10*3^{3}+10*3^{2}5*3^{1}+1 = +5*4^{4}10*4^{3}+10*4^{2}5*4^{1}+1 = 4^{5} And add the n:th general equation: +5*n^{4}10*n^{3}+10*n^{2}5*n^{1}+1= n^{5}
looking at the columns we see that 5*1^{4 }+ 5*2^{4 }+ 5*3^{4 }+ 5*4^{4 }+5*n^{4 }= +5S_{4} The second column is = 10S_{3} The third column is = +10S_{2} The fourth column is = 5S_{1} The fifth column is = S_{0} So we get that : 5S_{4 }10S_{3}+10S_{2}5S_{1}+S_{0 }= n^{5} Knowing the formulas for S2, S2, S1 and S0 we can get a simplified formula for S4: (Source: Mathologer in www.youtube.com ) 
NB The coefficient is inverse of the coefficient in the Pascal triangle rearranged in a matrix that is made inverse.
This is the last video I found about the Riemann hypothesis is from the Quanta Magazine channel:
He makes you dive through all the series studied by Euler to the Zeta function. It takes a while but it is worth it.
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The Ramanujan series and summation
There is a film about this Indian Mathematician Srinivasa Ramanujan:.
Srinivasa Ramanujan wrote to Hardy in England that
where the notation indicates a Ramanujan summation. (Wiki )
The Rieman zeta function comes to the same result.
It is hard to get make sense of 1+2+3+ … = 1/12 and some of those other notorious, crazylooking infinite sum identities.
Ramanujan said that he was inspired by a Hindu God called Namagiri He told Hardy about one of these revelations:
“While asleep, I had an unusual experience. There was a red screen formed by flowing blood, as it were. I was observing it. Suddenly a hand began to write on the screen. I became all attention. That hand wrote a number of elliptic integrals. They stuck to my mind. As soon
as I woke up, I committed
them to writing.” ( Wiki )
Hardy at Cambridge, wanted Ramanujan to prove his equations.
As I understand it Ramanujan was offended by this request as his God (Namagiri) does not share lies to a Hindu like Ramanujan. You can not question a God.
After some time, Ramanujan wrote however proof of the summation to Hardy. Thanks to Hardy’s insisting, we can now explain it.
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Ramanujan summation explained
I found Numberpiles video that explains how you can prove Ramajunan summation
I could take notes of the summations on a paper but I want to be able to read it here on my Number Theory page.
So I share a few print screens from Numberpiles video.
Numberphile uses two series S_{1} and S_{2} to get the solution of S
S_{1} = ½ as where you stop counting you will get either the sum 0 or 1.
We double S2
(OBS! you have to slide the second term of S_{2 }one step to right
as in this screenshot:
I have done that in my google drive calc and I got this table:
S1  Sum S1  S  4S  S_{2}  S_{2}  S2+S2  SS_{2} 
1  1  1  4  1  1  
−1  0  2  8  −2  1  −1  4 
1  1  3  12  3  −2  1  
−1  0  4  16  −4  3  −1  8 
1  1  5  20  5  −4  1  
−1  0  6  24  −6  5  −1  12 
We understand that S_{1} = ½
As 2S_{2}=S_{1 }then S_{2}= 1/4
We subtract S_{2} from S (=SS_{2})
By doing this, we get 4*S
As have seen that S_{2} = 1/4
This means that S – 1/4 = 4S. Subtracting 1S from both sides w get that – 1/4 = 3S
Dividing both sides with 3 we get that S = 1/12 !!!!
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Bernoulli numbers
Bernoulli numbers B_{n} are a sequence of rational numbers.
1/30 is one of the Bernoulli numbers. In fact the Bernoulli numbers are the coefficient for the nth coefficient in the series formulas like in this image below:
(Source: Mathologer in www.youtube.com )
The bernoulli numbers with the numbers from the Pascal triangle gives all the coefficients for the series like in the image below. Mathologer shows in his video minute 26:28 how.
(Source: Mathologer in www.youtube.com )
OBS. the numbers before the Bernoulli numbers are taken from the Pascal triangle. So with the Bernoulli numbers and the Pascal triangle you can write down any of the sums. Like
S_{10}= 1B_{0}n^{11} 12B_{1}n^{10} 66B_{2}n^{9} 220B_{3}n^{8} 495B_{4}n^{7} 792B_{5}n^{6} 924B_{6}n^{5} 792B_{7}n^{4} 49B_{8}n^{3} 220B_{9}n^{2} 66B_{10}n
I took out the coefficient creating a Pascal triangle in a Google Excel sheet
So the numbers that reappear in the result
(Source: Mathologer in www.youtube.com )
may have to do with the pascal triangle used in the sum equation.
The Bernoulli numbers appears in many ways like this
(Source: Mathologer in www.youtube.com )
And they appear also in the Riemann zeta function I wrote about in a separate page.
(Source: Mathologer in www.youtube.com )
2. Analytical NT
Analytical number theory uses analysis and complex analysis as a tool to address integer issues. Examples are the prime number and the related Riemann hypothesis . Other problems encountered with analytical methods are Waring’s problem , that a given integer represents a sum of squares, cubes, the prime numbering assumption, to find infinitely many prime pairs with the difference 2 and Goldbach’s premise, which suggests that even integers are the sum of two prime numbers.
Evidence that certain mathematical constants such as π and e are transcendent also belongs to analytical number theory. Statements about transcendent number appear to have moved from the study of integers. However, possible values of polynomials with integer coefficients for, for example, e, which are closely linked to the range of diophantic approximation, are studied .
Examples of methods used in analytical number theory are Hardy – Littlewood’s circular method , L functions and modular shapes .
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3. Algebraic NT
algebraic number theory is a branch of number theory that uses the techniques of abstract algebra to study the integers, rational numbers, and their generalizations. ( Wiki )
In algebraic number theory, the number area is expanded to include algebraic numbers, which are zeros to polynomials with coefficients that are integers. This set contains elements that are analogous to integers and are called algebraic integers . For these, wellknown properties, such as unique factorization, no longer need to apply. The tools employed are:
These tools give this number regions a partial order structure.
A large number of theoretical issues are addressed by studying the integers modulo p for all prime numbers p infinite bodies. This is called localization and leads to the construction of padic numbers . This type of study, which arose from algebraic number theory, is called local analysis .
An important area of algebraic number theory is the Iwasawa theory.
Diophantine equations may belong to this area too
“Diophantine equation, equation involving only sums, products, and powers in which all the constants are integers and the only solutions of interest are integers. For example, 3x + 7y = 1 or x^{2} − y^{2} = z^{3}, where x, y, and z are integers. Named in honour of the 3rdcentury Greek mathematician Diophantus of Alexandria, these equations were first systematically solved by Hindu mathematicians beginning with Aryabhata (c. 476–550).” . Read more at www.britannica.com
and in medium.com/cantorsparadise
Bernoulli numbers B_{n} are a sequence of rational numbers
Bernoulli could calculate easily the sum of
1^{10}+2^{10}+3^{10}….+1000^{10}
^{Mathologer makes a great introduction to numbers series and how to calculate these in this video https://www.youtube.com/watch?v=fw1kRz83Fj0}
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4. Geometry of NT
Geometric number theory encompasses all forms of geometry. It starts with Minkowski’s theorem,
The theorem was proved by Hermann Minkowski in 1889 and became the foundation of the branch of number theory called the geometry of numbers.
Minkowski’s theorem, is the statement that every convex set of which
 is symmetric with respect to the origin
 and which has volume greater than
contains a nonzero integer point.
The boundary of a convex set is always a convex curve. The intersection of all the convex sets that contain a given subset A of Euclidean space is called the convex hull of A. It is the smallest convex set containing A.
With Minkowsky’s theorem, one can also apply algebraic geometry , especially the theory behind elliptic curves . Fermat’s large proposition has been proven using these techniques.
The circle (black) intersects the line (purple) in two points (red). The disk (yellow) intersects the line in the line segment between the two red points. 

Convex set 
Non convex set  The red concave line segments contain the smallest convex set. 
Convex function
A function is convex if and only if its epigraph, the region (in green) above its graph (in blue), is a convex set. 
Primes and geometry
3 Blue 1 Brown shows a nice connection between primes, spirals and rays in this video:
Some images taken from this video:
Using prime numbers for radius () and angel in radius 

Each residue class from a spiral 
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1729, the Ramanujan number
It is a taxicab number, and is variously known as Ramanujan’s number and the Hardy–Ramanujan number, after an anecdote of the British mathematician G. H. Hardy when he visited Indian mathematician Srinivasa Ramanujan in hospital. (From the film “The Man Who Knew Infinity”.Hardy related their conversation:
I remember once going to see him when he was ill at Putney. I had ridden in taxi cab number 1729 and remarked that the number seemed to me rather a dull one, and that I hoped it was not an unfavourable omen. “No,” he replied, “it is a very interesting number; it is the smallest number expressible as the sum of two cubes in two different ways.” ( Wiki )
The two different ways are:
1729 = 1^{3} + 12^{3} = 9^{3} + 10^{3}
 1729 is
 1729 is also the third Carmichael number, the first Chernick–Carmichael number (sequence A033502 in the OEIS),
 the first absolute Euler pseudoprime.
 It is also a sphenic number.
 a Zeisel number.
 ^{} It is a centered cube number,
 ^{} a dodecagonal number,
 ^{}a 24gonal^{[10]} and 84gonal number.
 Schiemann found that such quadratic forms must be in four or more variables, and the least possible discriminant of a fourvariable pair is 1729.
 ^{} is the lowest number which can be represented by a Loeschian quadratic form a² + ab + b² in four different ways with a and b positive integers. The integer pairs (a,b) are (25,23), (32,15), (37,8) and (40,3). ( Wiki )
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5. Probabilistic NT
Probabilistic number theory addresses the likelihood of different number phenomena occurring within intervals or as relationships, such as the number of prime numbers within anumber interval.
This number theory has also led to the discovery of algorithms that Cramér’s assumption .
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6. Algorithmic number theory
In this area, algorithms are studied . Fast algorithms for prime number testing and integer factorization have widespread application in cryptography .
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Modular arithmetic NT
“Modular arithmetic is the part of number theory used to study cyclical systems: systems where taking enough steps “forward” brings you back to where you started.” (Brilliant.org)
(Source: Brilliant.org)
Modular arithmetic uses often special symbols. I have these linked here
congruence
“Image 1 is congruent (superimposable) only with image 2. Image 1 is similar but not congruent to image 3.”
In modular arithmetic the word congruence is used as in
“Given an integer n > 1, called a modulus, two integers are said to be congruent modulo n if n is a divisor of their difference, that is, if there is an integer k such that a − b = nk. Congruence modulo n is denoted:
In modulo 5 residue class
5k+3 Ξ 3, 8, 13,18, 23,,…
One can easily see that:
23 ≡ 3 (mod 5) as 23/5 =4 +3
23 ≡ 13 (mod 5) as 23/5=2+13
23 ≡ 18 (mod 5) as 23/5 =1+18
( WIki )
In the wheels below you find the congruent numbers in the same sector.
“in classic arithmetic, adding a positive number a to another number b always produces a number larger than b. In modular arithmetic this is not always so. For example, if it is now 4 o’clock and we “add” 23 hours, the time will then be 3 o’clock, which doesn’t appear to be larger than 4 o’clock.”(Source: www.math.upenn.edu/~mlazar/math170/notes06.pdf )
“A number is said to be congruent to 1 (modulo 4) if it leaves a remainder of 1 when divided by 4” ( www.math.brown.edu/~jhs/frintch1ch6.pdf )
Residue class modulo 4 is therefore = 4k+1 = 1,5,9,17,…
Modulo 5 wheel:
Residue classes modulo 5
5k+1 Ξ 1, 6, 11,16, 21,…
5k+n Ξ n, 5+n, 10+n,15+n, 20+n,…
The number in the blue section are all congruent to each other.
(Source: ZachStar video: www.youtube.com )
Modulo 7 wheel:
Residue classes modulo 7
7k+1 Ξ 1, 8, 15,16, 22,…
7k+n Ξ n, 7+n, 14+n,21+n, 28+n,…(Source: ZachStar video: www.youtube.com )
Modulo 7 wheel with conclusion:(Source: ZachStar video: www.youtube.com )
Modulo 12 wheel:
Residue classes modulo 12
12k+1 Ξ 1, 13, 25,37, 49,…
12k+n Ξ n, 12+n, 24+n,36+n, 48+n,…
Each section contains congruent numbers:
The number in the 5 section are all congruent to each other.
53/12=1 +41 ⇒ 53 Ξ 41 (modulo 12) or
41/12=1+29 ⇒ 41 Ξ 29 (modulo 12) or …
(Source: ZachStar video: www.youtube.com )
2 =14 in the clock. it is said that 2 is congruent ( ≡ ) to 14 modular 12
This can be written as 14 (mod12) ≡ 2
14/12 =1 +2
IMPORTANT.
the parentesis is not to be forgotten as if you write “14 mod 12” then it refers to the modulo operation used in computing lagnguages where you are looking to the rest of a division e.g. 15 mod 12 = rest 2.
Residue classes mod 6
(Source: 3Blue1Brown )
also these congruences are valid
14 (mod12) ≡ 2
26 (mod12) ≡ 2
38 (mod12) ≡ 2
50 (mod12) ≡ 2
14, 26,38, 50 are part of the congruence class of 2(mod 12)
(Source: www.youtube.com )
37 ≡ 57(mod 10)
The congruence relation may be rewritten as
because 38 − 14 = 24, which is a multiple of 12, or, equivalently, because both 38 and 14 have the same remainder 2 when divided by 12.
“modular arithmetic is a system of arithmetic for integers, where numbers “wrap around” when reaching a certain value, called the modulus. The modern approach to modular arithmetic was developed by Carl Friedrich Gauss in his book Disquisitiones Arithmeticae, published in 1801.
A familiar use of modular arithmetic is in the 12hour clock, in which the day is divided into two 12hour periods. If the time is 7:00 now, then hours later it will be 3:00. Simple addition would result in 7 + 8 = 15, but clocks “wrap around” every 12 hours. Because the hour number starts over after it reaches 12, this is arithmetic modulo 12. In terms of the definition below, 15 is congruent ( ≡ ) to 3 modulo 12, so “15:00” on a 24hour clock is displayed “3:00” on a 12hour clock.
For example,
The definition of congruence also applies to negative values:
” (Wiki )
I find this video is very interesting as it shows how modular arithmetic can be used.
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NT applications
Modular arithmetic in Cryptography
Zach start shares about cryptography with prime number and modular arithmetic in his ZachStar video: www.youtube.com from minute 16
In cryptography, modular arithmetic directly underpins public key systems such as RSA and Diffie–Hellman, and provides finite fields which underlie elliptic curves, and is used in a variety of symmetric key algorithms including
 Advanced Encryption Standard (AES),
 International Data Encryption Algorithm (IDEA),
 and RC4.
 RSA and Diffie–Hellman use modular exponentiation.(Wiki )
Modular arithmetic in banc account numbers
International Bank Account Numbers (IBANs), for example, make use of modulo 97 arithmetic to spot user input errors in bank account numbers. (Wiki )
In computer science
The logical operator XOR sums 2 bits, modulo 2 ( Wiki )
Sources
 http://www.numbertheory.org/ntw/N4.html
 https://www.math.brown.edu/~jhs/frintch1ch6.pdf
 https://brilliant.org/
 https://sv.wikipedia.org/wiki/Talteori (Swedish)
 https://en.wikipedia.org/wiki/Number_theory
 https://youtu.be/Qtl4nn7R4A
 http://www.math.umbc.edu/~campbell/NumbThy/Class/BasicNumbThy.html
 https://medium.com/
 Stackexchange
 medium.com/cantorsparadise/theriemannhypothesisexplainedfa01c1f75d3f
 Joseph H. Silverman, “What is number theory” )
 Think twice – animated math https://www.youtube.com/channel/UC9yt3wz6j19RwD5m5f6HSg
 About Gauss http://citeseerx.ist.psu.edu
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conclusion – ongoing process
I had a tough start with issue searching for a good introduction.
I then understood that this is a huge part of mathematics that will take a long time to explore and maybe my lifetime to understand.
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