"All of these books are excellent for learning number theory."
"We've proved that any 4k+1 prime can be written as a sum of two integers squares."
"Quaternions are an absolutely fascinating and often underappreciated number system from math."
"Euler 'outFermats' Fermat. His conjecture goes further and also asserts the impossibility of positive integer solutions to lots of closely related equations."
"Multiply a number by every number below it until it reaches 1."
"Negative numbers are funny in that sometimes they make sense sometimes they don't."
"Zero is an even number, I will die on this hill."
"This precise formula with all those complex and irrational numbers arranging themselves into the tribonacci numbers for all n is really a little miracle."
"Prime numbers: the building blocks of mathematics."
"Fermat primes and regular polygons: a remarkable connection."
"Ten includes all the other numbers; it is especially seven and three and is called the number of perfection."
"...any number to the fifth will retain its ones digit as mentioned in the beginning..."
"A natural logarithm is just a regular old logarithm. It's just that it has a very special number E for that base."
"Nobody knows whether or not there are infinitely many prime pairs."
"Any natural number can be factored uniquely into a product of prime powers."
"All natural numbers can be written as products of prime numbers."
"A common theme among my favorite results in number theory is that they arise from quite simple questions, but you discover something quite deep about the structure of mathematics."
"Every odd number that's large enough, if that beyond a certain point, every large odd number is the sum of three primes."
"So, Vinogradov's theorem tells you that if n is big enough, every odd number is the sum of three primes."
"We also know that every number less than 10 20 is the sum of three primes by complete different method."
"So, if you're counting patterns with two or more degrees of freedom, what the highly L conjecture is is proven for one for one parameter pattern like twins we we still it's it's still way off unfortunately."
"Primes are the fundamental building blocks of all numbers."
"Every prime squared is one bigger than a multiple of 24."
"...the quotient tells you the maximum number of divisors that is contained in this number or in the dividend."
"All natural numbers are either prime numbers or they can be expressed as a product of prime numbers."
"Small numbers can't do enough; there aren't enough small numbers to meet the many demands made of them."
"If you've got a product of two relatively prime numbers equaling a perfect cube, that means that each of them individually has to be a perfect cube."
"At their heart, all rings are generalizations of the integers."
"There's a huge amount of number theory that went into finding such a solution."
"It is meant to count primes or it's meant at least to find some simple curve or some simple function that's a reasonable approximation for the count of primes up to any number X."
"Prime numbers, square root accuracy, and spectrum."
"How many of them let PI of X be the number of primes less than or equal to X."
"This problem of solving these equations, as I said, is part of a greater problem that number theorists, mathematicians have worried about, solving equations in rationals and integers."
"It is impossible to divide a cube into two cubes, the fourth power into two fourth powers, or more generally any power higher than the fourth into two like powers."
"The equation X to the N plus Y to the N equals Z to the N has no solution in integers for n greater than or equal to 3 if X, Y, and Z are non-zero."
"The last 20 years has been really a very exciting time for number theorists."
"The Riemann hypothesis did inspire some other number theory in a way that's been very fruitful."
"N is the area of such a triangle if and only if the number of solutions of X, Y, Z in the integers such that 2X squared plus Y squared plus 8Z squared is N."
"If something's divisible by two coprime numbers, it's divisible by their product."
"That was one big step involved in the proof of the prime number theorem."
"That completes the proof of the prime number theorem."
"Express 56 as a product of prime factors."
"Therefore, one of n, n plus 1, or n plus 2 must be a multiple of three."
"Fermat's Little Theorem says that if p does not divide a, then a to the p minus one is congruent to one mod p."
"Integers can contain positive, negative whole numbers, and 0."
"Mathematics is the queen of the sciences, and number theory the queen of mathematics."
"Euler's totient function counts the number of natural numbers between 1 and n that are relatively prime to n."
"If the gcd of a with n is 1, then a to the phi of n is congruent to 1 mod n."
"R is a primitive root mod n if the order mod n of R is phi of n."
"Out of a primitive root mod p, we can create a primitive root mod p to the k for any natural number k."
"Yes, 126 is a term in this sequence and it's the 11th term."
"I certainly feel a bit of an obligation to do things to support the field of number theory and analytic number theory and mathematics more widely."
"The number twelve has way more factors than the number ten."
"Triangular numbers are the numbers that can be represented as a sum of consecutive integers starting with one."
"There isn't a biggest number, and whatever it is, you can always add one and get one bigger."
"I want to talk about what are called L-functions and the famous Riemann hypothesis."
"Every even number greater than 2 is the sum of two primes."
"Every even integer greater than two is the sum of two primes."
"A rational number is any number that can be written as a fraction of two integers."
"An Armstrong number is an n-digit number that is equal to the sum of the nth powers of its digits."
"It's a very interesting number, it's the smallest number expressible as a sum of two cubes in two different ways."
"N can be written as a sum of squares if and only if every prime of the form 4k plus 3 in the factorization of n occurs with an even exponent."
"The area that I work in personally is number theory, and for number theorists, we're interested in patterns of a specific kind, namely patterns in the universe of whole numbers."
"The prime factorization of 2020 is 2 squared times 5 times 101."
"Every positive whole number can be decomposed, can be expressed in a unique way as products of prime numbers."
"1729... it is the first number that's the sum of two cubes in two different ways."
"The greatest common divisor can always be written as a linear combination of the two integers in question."
"Every odd number greater than five can be written as the sum of three primes."
"Remember that zero, the number zero, is a multiple of every number."
"I really like this problem because it's a nice mix of electric circuits and also a little bit of number theory."
"This number is a special number in math or a number theory, this is called the golden ratio."
"Every even number higher than two is the sum of two prime numbers."
"All the whole numbers that you know are all rational numbers."
"The appearance of symmetry in the theory of numbers and as you see that there are some elements here which are kind of similar to what we saw in geometry."
"The ultimate goal of number theory or one of the ultimate goals is to understand what's called the algebraic closure of rational numbers."
"Show that 268 can be written as the sum of a power of 3 and a square number."
"If n is prime, then n minus 1 factorial is congruent to negative 1 mod n."
"If m and n can each be rewritten as the sum of two squares, so can m times n."
"A prime p can be rewritten as the sum of two squares if and only if p is equal to two or p is congruent to one mod four."
"It's a very beautiful idea that there has to be a particular number of ones and nines."
"If x squared is congruent to 1 mod p, then x is congruent to plus or minus 1 mod p."
"There are infinitely many primes that are congruent to one mod four."
"The probability that the GCD of two random numbers is one is related to pi. In fact, the probability that the GCD of number one and number two, two random numbers chosen, is six over pi squared."
"We will prove that there are infinitely many primes that are congruent to 3 mod eight."
"Every integer bigger than or equal to 2 can be written as a product of primes."
"These numbers are what we call pairwise relatively prime."
"This is number theory, which is really, really cool."
"The number 60 is a superior highly composite number; it has 12 factors."
"Every even number greater than 2 is the sum of two prime numbers."
"Znm is isomorphic to the direct product Zn cross Zm if and only if the greatest common divisor of n and m is equal to one."
"Every number can be represented as a sum of powers of twos."
"This is the definition of mod N that you'll find in number theory books."
"Number statements are assertions about concepts."
"The discovery of primes leads to the wonder, what are these special numbers? Leads to discovery of special properties of primes."
"We say that a and b are relatively prime if the greatest common divisor is actually equal to 1."
"If a number is divisible by two and three, it is a multiple of six."
"Wilson's theorem: that P minus 1 factorial is congruent to minus 1 mod p."
"Zeta of two, the sum of the reciprocal of the squares, is pi squared over 6."
"There exists a b such that a into b is congruent to one mod n."
"It takes all the multiples of a given number and sews together the integers from a line into a big loop, a big circle."
"Special numbers are defined as numbers less than 100 or greater than or equal to 300 that are perfectly divisible by 3, 7, or both."
"Euler finally settles this... he says, 'I know what this number is: it's pi squared over 6.'"
"This is the birth of analytic number theory."
"For a prime P equal to 4k plus 1, P must be equal to a sum of squares of integers."
"The greatest common divisor of two numbers is defined as the largest integer that divides both of them."
"The Euler totient function is the number of integers m between 1 and n that are relatively prime to n."
"The outer automorphism group is isomorphic to Z mod n star modded out by the subgroup plus or minus one."
"This theorem is telling the integer X has an inverse modulo M if and only if they're relatively prime."
"The real numbers often turn out to be the perfect number system in which to do math."
"The ABC conjecture says basically that if you have two numbers that add to a third, then C has to be at most a constant times all the primes dividing A, B, and C, just counted once."
"The density of square free numbers is actually equal to 6 over Pi squared."
"Next time we'll be looking at one of the oldest fields of mathematics in history, number theory."
"Integers a and B are congruent modulo some number M if a minus B is a multiple of M."
"If a divides b and a divides c, then a divides b plus c."
"The greatest common divisor of two numbers times the least common multiple of those two numbers is equal to the product of those numbers themselves."
"When the greatest common divisor of two nonzero integers A and B is only 1, then we say that A and B are relatively prime."
"We've managed to find the greatest common divisor of two numbers without ever finding any actual divisors of either number."
"Every even number greater than two can be represented as a sum of two prime numbers."