"The problem of writing primes in this way turns out to be at the heart of a ton of amazing and important mathematics."
"We've proved that any 4k+1 prime can be written as a sum of two integers squares."
"The actual definition of a prime number is it is a number with exactly two factors."
"Prime numbers are defined as whole numbers that are greater than one and are divisible by only one and themselves."
"Prime numbers are known as the building blocks of natural numbers."
"The SE finds its multiples of primes using multiplication, not by performing a division operation like trial division."
"Prime numbers are the building blocks of all numbers."
"Prime numbers are very mysterious and we know very little about them."
"Primes are perfectly structured in fact he heard the music of the primes."
"There is a perfect harmony, this is the perfect structure of the prime numbers."
"A lot of thought goes into generating random primes."
"Gauss's method only works for nth roots of unity where n is prime."
"These are the only prime-sided constructible polygons."
"A prime number can only be broken down into itself and one."
"Prime numbers like seven here are typically defined as whole numbers whose only positive factors are one and themselves whereas composite numbers have more than two positive factors."
"If a positive integer p is prime, then so is 2p + 1."
"...for any composite number x1 of its prime factors must be less than its own square root..."
"...any prime number squared besides two and three will land in this section..."
"Suppose you wanted to factorize a number of 2,000 bits into its primes..."
"The prime numbers are really a funny thing. I mean, are they random? Are they not random? They're sort of a mix of both. That's sort of what makes it so fascinating."
"Every prime number can be obtained in this way."
"The prime numbers exhibit stunning regularity."
"The smallest prime number that costs the same amount as the number beneath it."
"If we test a number with at least five different randomly chosen A's and they all say that n is prime, we can be 99.9% sure that n is really prime."
"The AKS test is the fastest known method for conclusively determining if any number is prime."
"The Great Internet Mersenne Prime Search (GIMPS) project has discovered the new largest known prime number 15 times."
"The biggest prime number we know has almost 25 million digits."
"Mathematicians have tried to shed light on the distribution of prime numbers."
"The reason why I'm interested what it means to study arithmetic is to study prime numbers but okay that seems like a random arbitrary thing but it's not the reason why I'm studying prime numbers is because they actually build up all of the whole numbers."
"The nth prime number is roughly a trillion times natural log of a trillion."
"The nth prime number is in log n. It's really quite weird."
"There's this large scale structure coming from Prim number theorem, there's this local what we call local structure."
"The primes are this sort of fascinating combination of three things."
"Every odd number that's large enough, if that beyond a certain point, every large odd number is the sum of three primes."
"We do know that there's infinitely many primes P such that p plus 2 is what we call an almost Prime."
"Primes are the fundamental building blocks of all numbers."
"For any prime, there will always be a bigger prime."
"Every prime squared is one bigger than a multiple of 24."
"All natural numbers are either prime numbers or they can be expressed as a product of prime numbers."
"Euclid proved that anyone who tried to write down the primes in some great big table would be writing forever because he proved that there are infinitely many primes."
"The primes never run out; there are infinitely many of these indivisible numbers."
"The Riemann hypothesis, if it's true, there's this pattern in the primes... it will explain to us why there aren't any patterns in the primes."
"A prime number is a positive integer that has exactly two different positive factors: one and itself."
"Two is the only even prime number."
"Two and three are the only consecutive numbers that are prime."
"Prime factorization gives you all the prime factors that a number can have."
"Trying to find where the next prime number is represents one of the biggest mysteries in the whole of mathematics."
"The twin prime conjecture is that there are infinitely many twin primes; you never run out."
"For me, the primes are like my hydrogen, helium, and lithium; they're the building blocks of the whole of mathematics."
"Prime numbers cannot be divided by any number other than 1 and the number itself without leaving a remainder."
"The highest prime number coalesced quietly in a corner and hid itself away forever."
"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."
"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."
"They're ordinary objects and they seem to sprout like weeds and nobody can predict where the next one will be."
"How many of them let PI of X be the number of primes less than or equal to X."
"The probability that a number is prime is proportional to its number of digits."
"It's approximated to PI of X is approximated by lye of X with essentially square root accuracy."
"You have a mystery quantity which is PI of X."
"The moral here is the Riemann spectrum holds the key to the position of prime numbers on the number line."
"13 and 17 just so happen to be prime numbers."
"The Prime Number Theorem is one of the most fundamental results in mathematics."
"If you can show that for every small epsilon, if x is large enough, then you have this estimate on the product of the primes up to x, then the theorem must hold."
"That was one big step involved in the proof of the prime number theorem."
"That completes the proof of the prime number theorem."
"Prime numbers... they have both an intrinsic regularity and a local irregularity."
"...every deep question on primes would be solved if we know the Riemann hypothesis."
"The primes are ornery, seeming to obey no other law than that of chance."
"Euclid proved that there are infinitely many primes."
"I do believe that the twin primes conjecture is true, I do believe we will manage to prove it."
"There are infinitely many primes. There is no biggest prime number."
"The world is a much more interesting place with infinitely many primes."
"Apart from 2 & 3, every prime in the world is 1 less than a multiple of 6 or 1 more than a multiple of 6."
"You can use the following Python code to check if a number is a prime number."
"The prime numbers grow like weed among the natural numbers."
"Primes are numbers only divisible by 1 and themselves."
"The ancient Greeks already knew there are infinitely many primes."
"If among the first n numbers you pick one at random, the probability of picking a prime number is approximately 1 over the logarithm of n."
"God may not play dice with the Universe, but there's something strange going on with the prime numbers."
"Prime numbers are numbers divisible only by themselves and one, and we know there are infinitely many of those."
"Every prime number of the form 4k + 1 can be written as the sum of two squares."
"Normal Sudoku rules apply and digits in cages cannot repeat and must sum to a prime number."
"For the avoidance of doubt, one is not a prime number."
"The purpose of this program is to compute the sum of the first n prime numbers."
"If it's not divisible by any of these, chances are it's prime."
"A number is prime if it's only divisible by 1 and itself."
"Euler's totient function counts the number of natural numbers between 1 and n that are relatively prime to n."
"There is a primitive root mod p for every prime."
"7 is prime in the set of integers because it cannot be factored into a product of smaller numbers in Z."
"Eisenstein's criterion says that if we can find a prime that divides all of the coefficients of the polynomial except for the leading coefficient, then we can conclude that our polynomial is irreducible."
"A prime is an integer greater than one not divisible by anything except one and itself."
"Prove by contradiction that there are infinitely many prime numbers."
"This is a contradiction, therefore, there must be an infinite number of primes."
"It's really important that the word 'smallest' is here because perhaps there are a bunch of primes that can be written in this form, but we want the smallest one."
"If we have an odd prime p and any reduced residue system modulo p, there are exactly p minus one over two quadratic residues and p minus one over two quadratic non-residues."
"A prime number is a number that is divisible by one and itself; it only has two factors."
"73 is the 21st prime number, its mirror, 37, is the 12th, and its mirror, 21, is the product of multiplying 7 and 3."
"Our universe is the universe of whole numbers and prime numbers form the atoms of this universe."
"Prime numbers form the atoms of our universe."
"Prime numbers are the atoms of our universe of the whole numbers."
"Prime numbers play a role in our lives every day."
"The fact that it's easy to multiply two prime numbers together but hard to do the reverse process, that's the basis of much of the world's encryption schemes."
"What's wrong with prime numbers? They're impossible to divide up evenly."
"There are an infinite number of prime numbers separated by just two."
"We're making sense of something—everybody can understand why three is a prime number, why five is a prime number."
"A prime number is any number that's only divisible by 1 and itself."
"If I've got two numbers that multiply together to give you that number, then one of them has to be one, otherwise it's not a prime."
"If n is prime, then n minus 1 factorial is congruent to negative 1 mod n."
"If a prime can be expressed as the sum of two squares, then that expression is unique."
"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."
"The Riemann Zeta function is a mathematical object, a surface which is designed to understand the prime numbers."
"If you understand the behavior of the Riemann Zeta function, you can deconstruct it to get information about the prime numbers."
"The Riemann Hypothesis is that these special points where all the colors meet lie on a single vertical straight line."
"Each cage contains a prime number, read from left to right."
"We will prove that there are infinitely many primes that are congruent to 3 mod eight."
"So there must be infinitely many primes."
"For n equals two, well, two itself is prime."
"These numbers are what we call pairwise relatively prime."
"Therefore, we have a contradiction and there must be infinitely many prime numbers."
"The least common multiple of n and m is nm when n and m are relatively prime."
"Euclid's proof of infinitely many primes is a mathematical contradiction you're going to have to remember."
"All of that stuff is based on mathematics; in fact, it's based on prime numbers and factorization."
"A prime number is a number which can only be divided evenly by the number one and itself."
"The discovery of primes leads to the wonder, what are these special numbers? Leads to discovery of special properties of primes."
"The first prime number from the list of sixteen we've created is three."
"The Riemann Hypothesis, an open conjecture related to prime numbers, has implications for their distribution."
"Meren primes are prime numbers that have the property of being 2 to the N minus one where N is prime."
"Prime numbers are very important to the whole encryption process."
"The greater the number of primes that we can decompose N in, and the smaller each of those primes, the higher the efficiency of the fast Fourier transform algorithm will be."
"Prime numbers are numbers that are only divisible by themselves and one."
"Your prime numbers are numbers that are only divisible by themselves and one."
"There always exists a subgroup capital H which is a subgroup of capital G such that the order of capital H is equal to P I to the power of alpha I for any I that you like between 1 and R."
"A prime are the numbers 1, 3, 5, and 7."
"Prime numbers are a kind of atoms which composed together to make all the numbers you can think of them as building blocks."
"There exist infinitely many primes."
"It's really an amazing result no matter what a is no matter which prime n we choose."
"We have solar panels arranged in prime number sequences in the same orbit; there's no way you would mistake this for a planet."
"Prime numbers have two factors: one and themselves."
"Every cell that is in a prime numbered position must contain a prime digit."
"Let n be a prime... then we have seen if a is relatively prime with n, that means gcd of a and n is one."
"Prime numbers exclude zero and one from being prime by definition."
"Let me give you an example of some prime numbers: the number two is a prime number."
"Euler's product formula is not only useful since it gives us another way to write the Zeta function, but it also connects the Zeta function to the prime numbers."
"There exists infinitely many primes."
"Any field at all, its characteristic, if it's not zero, has got to be a prime number."
"If P equals a squared plus b squared, then a plus or minus bi are primes in the Gaussian integers."
"For a prime P equal to 4k plus 1, P must be equal to a sum of squares of integers."
"A prime number means they only have two factors: one and themselves."
"Prime numbers mean they only have two factors: one and themselves."
"To eliminate all terms of the form one over a multiple of prime p, we multiply our series by one plus or minus one over p."
"Pi over four is the product of all odd primes divided by themselves rounded to the nearest multiple of four."
"The product of the odd primes divided by their nearest multiples of four is pi over four."
"A prime number is only divisible by one and itself."
"The prime numbers are the building blocks of the mathematical world."
"If n is prime, then any element other than zero in Jp minus zero is a primitive element or is a generator."
"Choose big enough primes, and even the fastest computers today would take more than the life of the entire universe to factor their product."
"You can see right there immediately just how good of an idea it is to use prime numbers."
"The prime numbers can be thought of as the building blocks on which the entire set of natural numbers is built."
"The quest for larger and larger Mersenne Primes has been carried out by what is known as the great internet Mersenne prime search."
"If a number is not divisible by any of the numbers up to its square root, then it's a prime number."