- He developed the idea of numerical logic and was responsible for the first golden age of mathematics. He studied the properties of particular numbers, the relationships between them, and the patterns they formed. He realized that numbers exist independently of the tangible world. He traveled and gathered many mathematical techniques in the world. He made numbers no longer merely used to count and calculate, but were appreciated in their own right.
- He and his followers studied “excessive” number, “perfect” number and “twoness”. (p11)
- He discovered the fundamental relationship between the harmony of music and the harmony of numbers, and the relationship between mathematics and science. (p14+17)
- Pythagoras’s theorem: In a right-angled triangle the square on the hypotenuse is equal to the sum of the squares on the other two sides. ( x
^{2}+ y^{2}= z^{2}). (p19)

- He founded no fraction for √2 existed. His master, Pythagoras, who defined the universe in terms of rational numbers, was unwilling to accept that he was wrong. He denied irrational numbers, which was a disgraceful act.

- He wrote the most successful textbook in history “Elements”.
- He exploited a logical weapon “reductio ad absurdum” (proof by contradiction): trying to prove that a theorem is true by first assuming that the theorem is false
- He established the existence of “irrational numbers”, which developed the abstract quality of numbers
- He proved that √2 couldn’t be written as a fraction
- He examined that √2 multiplied by itself is equal to “2”.
- He discovered that perfect numbers are always the multiple of two numbers, one of which is a power of 2 and the other being the next power of 2 minus 1 (although today there are enormously large examples founded that obeys Euclid’s rule).
- He proved that there is an infinity of prime numbers. Initially he assumed that there was a finite list of known prime numbers, and then he showed that there must exist an infinite number of additions to this list. There are N prime numbers in Euclid’s finite list, which are labeled P1, P2, P3, …, PN. Then he could generate a new number QA such that: QA = (P1 × P2 × P3 × ... × PN ) + 1
- He discovered “Unique factorization”, which states that there is only one possible combination of primes that will multiply together to give any particular number ex: 18=2×3×3, 35=5×7, 180=2×2×3×5×7×11×23 (p46-50, p91, p114)

- Archimedes invented 'integral calculus'. Using this, he measured the section of areas surrounded by geometric figures. He broke the sections into a number of rectangles and then added the areas together.
- He discovered the value of √3 is more than 265/153 (approximately 1.732) and less than 1351/780 (approxiately 1.7320512). The accurate value is rounded to 1.7320508076.
- In 'The Quadrature of the Parabola', he proved that the area enclosed by a parabola and a straight line is 4/3 multiplied by the area of a triangle with equal base and height. http://www.ancientgreece.com/s/People/Archimedes/

- He studied elliptic equations/elliptic curves, which states that y
^{2}= x^{3}+ ax^{2}+ bx + c, where a, b, c are any whole numbers. By simply changing the values of a, b, and c in the general elliptic equation mathematicians can generate an infinite variety of equations.

- He was the first person to use division by zero as a definition for infinity.

- He discovered imaginary number, "i", in the sixteenth century (1572), contributed to the complex number theory. (p81+84)

- He was the first person who proposed that each point in two dimensions can be described by two numbers on a plane, one giving the point’s horizontal location, and the other the vertical location, which have come to be known as Cartesian coordinates.

- He was the first one to claim that the reason why nobody could find any solutions of x
^{2}+ y^{2}= z^{2}for n greater than 2 was that there was no solution existed. However, his proof had been lost long time ago. (p31) - He and Pascal discovered the essential mathematical rules that more accurately describe the laws of chance. These rules can be applied in many fields such as speculating on the stock market and estimating the probability of a nuclear accident. (p41-43)
- He was deeply involved in the founding of calculus, which helps revolutionize
- He discovered “friendly numbers”, which are pairs of numbers such that each number is the sum of the divisors of the other number (p58)
- He declared “It is impossible for a cube to be written as a sum of two cubes or a fourth power to be written as the sum of two fourth powers or, in general, for any number which is a power greater than the second to be written as a sum of two like powers.” (p61)

- As Fermat’s eldest son, he allowed people in the world to know about Fermat’s remarkable breakthroughs in number theory. Without him, Fermat’s Last Theorem would have died with its creator.

- He stated that “the excitement that a gambler feels when making a bet is equal to the amount he might win multiplied by the probability of winning it”. (p43) Studying on the subject of gambling problems helped create the mathematical theory of probabilities.
- He created the famous and widely-used "Pascal's Triangle". https://en.wikipedia.org/wiki/Blaise_Pascal#Contributions_to_mathematics

- He applied mathematics to the physical world, helping change the culture of numbers. (p74)

- He proved Fermat’s prime theorem in 1749, which stated that the first type of primes were always the sum of two squares ( 13 = 2
^{2}+ 3^{3}), whereas the second type could never be written in this way. (P63) - He developed “algorithm”, which can generate an imperfect but sufficiently accurate solution (p76)
- He claimed that he had an algebraic proof for the existence of God: ( a+b
^{n}) / n = x (p76) - He used Fermat’s method to prove the case for n=3 in 1753. (p81)

- She claimed that it was unlikely that any solutions existed for the equation x
^{n}+ y^{n}= z^{n}because if there was a solution then either x, y, or z would be a multiple of n. (p106)

- In 1843, he showed Dirichlet an attempted proof of Fermat’s last theorem. Dirichlet found an error, and Kummer continued his search and developed the concept of ideal numbers. He proved the insolubility of the equation for all but a small group of primes, and thereby he laid the foundation for an eventual complete proof of Fermat’s last theorem. https://www.britannica.com/biography/Ernst-Eduard-Kummer

- He classified all quintics into two types: those that were soluble and those that were not. Then for those that were soluble, he devised a recipe for finding the solutions to the equations. Moreover, he examined and identified the soluble equations for higher order than the quintic, those containing x
^{6}, x^{7}, and so on. His finding was one of the masterpieces of nineteenth century mathematics. (p227)

- Hilbert’s Hotel: suggests that all infinities are as large as each other, because various infinities seem to be able to squeeze into the same infinite hotel.

- In 1940, he declared that the best mathematics is largely useless. It has no effect on war. (p147)

- He was the first person who suggested that rivers have a tendency toward an ever more loopy path because the slightest curve will lead to faster currents on the outer side, which will in turn result in more erosion and a sharper bend.

- He cowrote the book "The Theory of Games and Economic Behavior", using mathematics to describe the structure of games and how human play them. (p147)

- He created two theorems of undecidability:
- “First Theorem of Undecidability”: If axiomatic set theory is consistent, there exist theorems that can neither be proved or disproved. (No matter what set of axioms were being used there would be questions that mathematics could no answer -- completeness could never be achieved.)
- “Second Theorem of Undecidability”: There is no constructive procedure that will prove axiomatic theory to be consistent. (Mathematicians could never even be sure that their choice of axioms would not lead to a contradiction -- consistency could never be proved.)
- His work on undecidability had introduced an element of doubt – whether Fermat’s Last Theorem was soluble. (p139-147)

- During the the Second World War, he contributed most to the code-cracking effort. When the war ended, he continued to build increasingly complex machines, such as the Automatic Computing Engine (ACE). in 1948, he built the world’s first computer to have an electronically stored program. (p156)

- He contributed to the Taniyama–Shimura conjecture, which states that elliptic curves over the field of rational numbers are related to modular forms.

- He contributed to the Taniyama–Shimura conjecture, which states that elliptic curves over the field of rational numbers are related to modular forms.
- He claimed that every elliptic equation can be linked to modular forms although he could not prove that it was true. This conjecture has potential to firstly connect two areas of mathematical study together.

- He changed Fermat’s Last Theorem into A
^{N}+ B^{N}= C^{N.}Then he rearranged the equation into y^{2}= x^{3}+ ( A^{N}- B^{N}) x^{2}- A^{N}B^{N}, which is an elliptic equation. - He also had argument:
- If (and only if) Fermat’s Last Theorem is wrong, then Frey’s elliptic equation exists.
- Frey’s elliptic equation is so weird that it can never be modular.
- The Taniyama-Shimura conjecture claims that every elliptic equation must be modular.
- Therefore the Taniyama-Shimura conjecture must be false!

- Then he ran his argument backward, and concluded that if mathematicians could prove the Taniyama-Shimura conjecture then they would automatically prove Fermat’s Last Theorem. This led mathematicians to try to prove that an elliptic equation is not modular. (p195-197)

- Inspired by Gerhard Frey, he proved that Frey’s elliptic equation is not modular, completing the connection between the Taniyama-Shimura conjecture and Fermat’s Last Theorem. (p199+202)

- She claimed to have a proof for Fermat’s Last Theorem using geometric approach. But a part of her logic was discovered to be wrong later.

- He was the first person who proved Fermat's Last Theorem 1994.

- He was one of the referees responsible for verifying Wiles’s proof and a former student of Wiles. Wiles decided to invite him to work alongside him.

- He claimed that he had a counterexample of Fermat’s Last Theorem.

Counterexample: 2,682,440^{4}+ 15,365,639^{4}+ 18,796,760^{4}= 20,615,673^{4}But this is not exactly Fermat’s Last Theorem since there are three numbers added up instead of two.

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