Logic Math & Sciences - Number theory

Number theory

Updated 1/1/2026)

Prime Numbers

Definition

Prime Numbers in Nature

Ordinals and Cardinals

Interesting Numbers:

The first 10,000 Primes

The first 23 Perfect Numbers

The Number Pi

The Evolution of Numbers

The concept of "number" has evolved philosophically from simple arithmetic to complex operations. We can view this hierarchy like Russian nesting dolls: generally, each new category encompasses the previous one to solve problems the simpler numbers could not.

N = Natural Numbers {0, 1, 2, ...}. These are the ancient, intuitive "counting numbers."
Z = Integers {...., -1, 0, 1, ....}. This set expands the Natural Numbers to include their negative counterparts.
Q = Rational Numbers. Any number expressible as a ratio of two integers. This set includes all Integers.
Irrational Numbers. Here the "nesting" pauses. These are numbers that cannot be expressed as a ratio of integers (e.g., pi, or √2. They are distinct from Rational numbers.
Algebraic Numbers. Any number that solves a polynomial equation with integer coefficients (e.g., x²- 2 = 0). This includes all Rational numbers and some Irrationals like √2.
Transcendental Numbers. Numbers that are not solutions to any algebraic equation. They "transcend" algebra. Famous examples include pi and e.
R = Real Numbers. The complete union of all Rational, Algebraic, and Transcendental numbers. This represents the continuous line without gaps.
C = Complex Numbers. These introduce "imaginary" numbers required to solve problems involving the square roots of negative numbers. This set includes all Real numbers.
Infinite Numbers. These express differing cardinalities (sizes) for infinite sets of numbers and sit distinct from the algebraic sets above.

Prime Numbers

Prime numbers are the basic units of the system of numbers since every natural number is either a prime or can be expressed as a composite of prime numbers. Here are some excerpts from the Wikipedia article about Prime numbers:

Definition

"A prime number (or a prime) is a natural number greater than 1 that has no positive divisors other than 1 and itself. A natural number greater than 1 that is not a prime number is called a composite number. For example, 5 is prime, as only 1 and 5 divide it, whereas 6 is composite since it has the divisors 2 and 3 in addition to 1 and 6. The fundamental theorem of arithmetic establishes the central role of primes in number theory: any integer greater than 1 can be expressed as a product of primes that is unique up to ordering. This theorem requires excluding 1 as a prime.

The property of being prime is called primality. A simple but slow method of verifying the primality of a given number n is known as trial division. It consists of testing whether n is a multiple of any integer between 2 and √n. Algorithms that are much more efficient than trial division have been devised to test the primality of large numbers. Particularly fast methods are available for primes of special forms, such as Mersenne primes. As of 2011, the largest known prime number has nearly 13 million decimal digits.

There are infinitely many primes, as demonstrated by Euclid around 300 BC. There is no known useful formula that yields all of the prime numbers and no composites. However, the distribution of primes, that is to say, the statistical behavior of primes in the large can be modeled. The first result in that direction is the prime number theorem, proven at the end of the 19th century, which says that the probability that a given, randomly chosen number n is prime is inversely proportional to its number of digits, or the logarithm of n.

Many questions around prime numbers remain open, such as Goldbach's conjecture, which asserts that every even integer greater than 2 can be expressed as the sum of two primes, and the twin prime conjecture, which says that there are infinitely many pairs of primes whose difference is 2. Such questions spurred the development of various branches of number theory, focusing on analytic or algebraic aspects of numbers. Primes are used in several routines in information technology, such as public-key cryptography, which makes use of properties such as the difficulty of factoring large numbers into their prime factors. Prime numbers give rise to various generalizations in other mathematical domains, mainly algebra, such as prime elements and prime ideals."

Prime Numbers in Nature

“Inevitably, some of the numbers that occur in nature are prime. There are, however, relatively few examples of numbers that appear in nature because they are prime.

One example of the use of prime numbers in nature is as an evolutionary strategy used by cicadas of the genus Magicicada. These insects spend most of their lives as grubs underground. They only pupate and then emerge from their burrows after 13 or 17 years, at which point they fly about, breed, and then die after a few weeks at most. The logic for this is believed to be that the prime number intervals between emergences make it very difficult for predators to evolve that could specialize as predators on Magicicadas. If Magicicadas appeared at non-prime number intervals, say every 12 years, then predators appearing every 2, 3, 4, 6, or 12 years would be sure to meet them. Over a 200-year period, average predator populations during hypothetical outbreaks of 14- and 15-year cicadas would be up to 2% higher than during outbreaks of 13- and 17-year cicadas. Though small, this advantage appears to have been enough to drive natural selection in favor of a prime-numbered life-cycle for these insects.

Also: There is speculation that the zeros of the zeta function are connected to the energy levels of complex quantum systems.“

Prime Number Theorem

The Prime Number Theorem (PNT) describes the distribution of prime numbers. Euclid proved that there is an infinite number of primes, but their location can only be predicted by statistical means, as an approximation. As ordinary numbers get larger and larger, we find fewer and fewer prime numbers scattered among them. This can easily be demonstrated with the "Sieve of Eratosthenes." It has resulted in a hunt to find a formula for the location of primes, which is also essential for encrypting messages sent across public information highways. Even though we cannot predict primes (yet?), we find that the distribution of prime numbers is asymptotic, and follows very precise laws. In simple terms, the Prime Number Theorem says that, as x approaches infinity, the quotient between x and ln(x) becomes a better and better estimate of the number of primes at or below x. That's not a lot, but it is better than just saying that primes are distributed randomly. There is a pattern to their distribution, and there is still a lot more to find out.

Five interesting Things about Mersenne Primes
New Pattern found in Prime Numbers (from Phys.org)
An elegant proof from Erdős
Conspiracy of the Primes: Wired Magazine, 2016.

Perfect Numbers

Perfect numbers are gateways to the wonders of the mathematical world. Contemplating them, one realizes how small our human minds really are compared to the reality that surrounds us, and creates us.

Definition

A perfect number is a positive integer that is equal to the sum of its positive divisors excluding itself. The first perfect number is six. Six is the number of sides to each cell in the bee's honeycomb, or the number of points of all snowflakes. ALL SNOWFLEKES HAVE SIX CORNERS, OR POINTS. Amazing, isn't it? There is an infinite number of snowflakes, each different from every other, but every one of them has six points.

Six is a perfect number because both the sum and the product of its constituents add up to six: 1+2+3=6, or 1 x 2 x 3=6.

The second perfect number is 28: 1+2+4+7+14=28. This is followed by the perfect numbers 496 and 8128.

( A clear and in-depth discussion of the relationship between primes and perfect numbers, as well as the history of mathematical discovery, can be found at the MacTutor site, written by J J O'Connor and E F Robertson. )

Discovery

Greek Mathematicians only knew about the first four perfect numbers; the mathematician Nicomachus had noted 8,128 in the first century CE. In 1456, someone recorded a fifth perfect number, 33,550,336. In 1588, the Italian mathematician Pietro Cataldi found the sixth (8,589,869,056) and the seventh (137,438,691,328) perfect numbers.

Euclid found a formula that works for the first four perfect numbers: 2p−1(2p−1), (p is a prime number):

for p = 2: 21(22−1) = 6

for p = 3: 22(23−1) = 28

for p = 5: 24(25−1) = 496

for p = 7: 26(27−1) = 8128.

He proved that whenever 2p−1 is a prime number, 2p−1(2p−1) is an even perfect number.

For 2p−1 to be prime, it is necessary that p itself be prime. Prime numbers of the form 2p−1 are so-called "Mersenne primes". after the seventeenth-century mathematician Marin Mersenne.

Not all numbers of the form 2p−1 with a prime p are prime numbers. For example, 211−1 = 2047 = 23 × 89. In fact, Mersenne primes are very rare—of the 78,498 prime numbers p below 1,000,000, 2p−1 is prime for only 33 of them.

In the 18th century Leonhard Euler proved that the formula 2p−1(2p−1) will yield all the even perfect numbers. He found that there is a one-to-one relationship between even perfect numbers and Mersenne primes; each Mersenne prime generates one even perfect number, and vice versa. This result is called the Euclid–Euler Theorem. As of June 2010, we know 47 Mersenne primes and therefore 47 even perfect numbers. The largest of these is 243,112,608 × (243,112,609−1) with 25,956,377 digits.

There is no proof that there are infinitely many Mersenne primes and perfect numbers.

Fibonacci Sequence

Definition

A Fibonacci sequence is easily constructed: Start with 0 and 1, and for each following number, add the previous two: 0, 1, 0+1=1, 1+1=2, 1+2=3, and so on. Here is the beginning of the sequence: 0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, ...

The Fibonacci sequence is named after Leonardo of Pisa, who is also known as Fibonacci. In 1202 he wrote a book entitled "Liber Abaci", which introduces the sequence to Western European mathematics. It had been described earlier, however, in Indian mathematics.

Fibonacci number sequences are everywhere in nature, and the math can quickly get very complex. Some numbers in the sequence are also prime, which creates an interesting way to look at primes, and more problems, for instance: Are there an infinite number of Fibonacci Primes?

Here is some math from the Wikipedia:

Fibonacci primes

A Fibonacci prime is a Fibonacci number that is prime. The first few are:

2, 3, 5, 13, 89, 233, 1597, 28657, 514229, 433494437, 2971215073

Fibonacci primes with thousands of digits have been found, but it is not known whether there are infinitely many.

Fkn is divisible by Fn, so, apart from F4 = 3, any Fibonacci prime must have a prime index. As there are arbitrarily long runs of composite numbers, there are therefore also arbitrarily long runs of composite Fibonacci numbers.

With the exceptions of 1, 8 and 144 (F1 = F2, F6 and F12) every Fibonacci number has a prime factor that is not a factor of any smaller Fibonacci number (Carmichael's theorem).

144 is the only nontrivial square Fibonacci number. Attila Pethő proved in 2001 that there are only finitely many perfect power Fibonacci numbers. In 2006, Y. Bugeaud, M. Mignotte, and S. Siksek proved that only 8 and 144 are non-trivial perfect powers.

Surprising applications of math (johndcook.com)
Fibonacci: the man who figured out flowers (telegraph.co.uk)
Nature by the numbers: Fibonacci Sequence animated in mesmerizing video (geek.com)

Ordinals and Cardinals

How do you represent numbers in language?

(Quoted from: Pi in the Sky, by John Barrow. Oxford University Press, 1992. p. 37-38)

"English: one/first ; two/second ; three/third ; four/fourth
French: un/premier ; deux/second or deuxième ; trois/troisième ; quatre/quatrième
German: ein/erste ; zwei/ander or zweite ; drei/dritter ; vier/vierte
Italian: uno/primo ; due/secondo ; tre/terzo ; quattro/quarto

In each of these four languages the words for 'one' and 'first' are quite distinct in form and emphasize the distinction between solitariness (one) and priority (being first). In Italian and the more old-fashioned German and French usage of ander and second, there is also a clear difference between the words used for 'two' and 'second', just as there is in English. This reflects the Latin root sense in English, French, and Italian of being second, this is, coming next in line, and this does not necessarily have an immediate association with two quantities. But when we get to three and beyond, there is a clear and simple relationship between the cardinal and ordinal words. Presumably this indicates that the dual aspect of number was appreciated by the time the concepts of 'threeness' and 'fourness' had emerged linguistically, following a period when only words describing 'oneness' and 'twoness' existed with greater quantities described by joining those words together as we described above.

In all the known languages of Indo-European origin, numbers larger than four are never treated as adjectives, changing their form according to the thing they are describing. But, numbers up to and including four are: we say they are 'inflected'. [...] a rather antiquated structure that barely survives in the modern forms of many Indo-European languages. For example, in French we find two words un and une corresponding to the English 'one' and they are used according to the gender of what is being counted. An analogous feature of language that certainly survives in English is the way in which different adjectives are associated with the same quantities of different things. We speak of a pair of shoes, a brace of pheasants, a yoke of oxen, or a couple of people, but we would never speak of a brace of chickens or a couple of shoes. [...]

We have seen that the distinction between cardinal and ordinal aspects of number and the use of inflected adjectives is clear up to the number four but conflated beyond that. [Footnote: In Finnish there are still two kinds of plural, as in classical Greek, Biblical Hebrew and Arabic: one for two things and another for more than two. Also interesting in this respect is the fact that there is no connection between the words for '2' and '½' in the Romance and Slavic languages (nor in Hungarian which is not an Indo-European language) but in all the European languages the words for '3' and '1/3', '4' and '1/4' and so on, are closely related, just as they are in English. This may indicate that the concept of a fraction, or the relation between a number and the concept of a ratio, only emerged after counting beyond 'two'.]

[...]

A curious speculation arises [...] to give special status to the number 8 - the total number of fingers excluding the thumbs - that many known languages originally possessed a base-8 system (which they later replaced by something better), because the word for the number 'nine' appears closely related to the word for 'new' suggesting that nine was a new number added to a traditional system. There are about twenty examples of this link, including Sanskrit, Persian, and the more familiar Latin, where we can see novus = 'new' and novem = 'nine'."