Proof in Mathematics
Four colour theorem
The problem with maps; Kenneth Appel
Dr Appel's conjecture had to do with maps. For more than a century, ever since Francis Guthrie, a maths student, had had a stray thought in 1852, mathematicians had been wondering how many colours it would take to ensure that, no matter how complex a map, no adjacent countries would be the same colour. Guthrie thought the answer was four, but it was just an intuition. He could not prove it mathematically, and nor, for decades, could anyone else. It was a matter of constantly creating new configurations, or maps, and proving that, for them, the four-colour conjecture was true.
"The problem with maps; Kenneth Appel." The Economist, vol. 407, no. 8834, 4 May 2013, p. 90(US). Gale In Context: High School, link.gale.com/apps/doc/A328394192/SUIC?u=61_alls&sid=bookmark-SUIC&xid=16f8022a. Accessed 2 July 2021.
Mordell and Faltings
The Mordell Conjecture / Falting's Theorem
The Mordell conjecture states that Diophantine equations that give rise to surfaces with two or more holes have only finite many solutions in Gaussian integers with no common factors (Mordell 1922). Fermat's equation has 
holes, so the Mordell conjecture implies that for each integer 
, the Fermat equation has at most a finite number of solutions.
This conjecture was proved by Faltings (1984) and hence is now also known as Falting's theorem.
Weisstein, Eric W. "Mordell Conjecture." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/MordellConjecture.html
German mathematician Gerd Faltings (1954-) proved Mordell's conjecture in 1983, an accomplishment that earned him the prestigious Fields Medal, mathematics' highest honor. His method of altering a familiar geometric theorem into algebraic terms led him to solve the complex geometric theorem proposed by Louis Mordell in 1922. His success in proving this conjecture has contributed to the advancement of the studies of algebra and geometry.
Gerd Faltings Proves Mordell's Conjecture (1983). (2001). In N. Schlager & J. Lauer (Eds.), Science and Its Times (Vol. 7). Gale. https://link.gale.com/apps/doc/CV2643450805/SUIC?u=61_alls&sid=bookmark-SUIC&xid=d17fa50d
Gerd Faltings, 2005.
Renate Schmid—Mathematisches Forschungsinstitut Oberwolfach gGmbH/Oberwolfach Photo Collection (MFO Photo ID: 7513) Creative Commons Attribution ShareAlike 2.0 (Germany)
Gerd Falting's proof was a major breakthrough in proving Fermat's Last Theorem that this equation has no natural number solutions for n > 2.
Gerd Faltings. (2000). In N. Schlager & J. Lauer (Eds.), Science and Its Times (Vol. 7). Gale. https://link.gale.com/apps/doc/K2643413067/SUIC?u=61_alls&sid=bookmark-SUIC&xid=60ed87c8
Fermat's Last Theorem
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Search term - Fermat's last theorem
Fermat’s last theorem, also called Fermat’s great theorem, the statement that there are no natural numbers (1, 2, 3,…) x, y, and z such that xn + yn = zn, in which n is a natural number greater than 2. For example, if n = 3, Fermat’s last theorem states that no natural numbers x, y, and z exist such that x3 + y 3 = z3 (i.e., the sum of two cubes is not a cube).
"Fermat’s last theorem." Britannica School, Encyclopædia Britannica, 9 Mar. 2017. school.eb.com.au/levels/high/article/Fermats-last-theorem/34050. Accessed 1 Jul. 2021.
Goldbach's Conjecture
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Search term - Goldbach conjecture
Goldbach conjecture, in number theory, assertion (here stated in modern terms) that every even counting number greater than 2 is equal to the sum of two prime numbers. The Russian mathematician Christian Goldbach first proposed this conjecture in a letter to the Swiss mathematician Leonhard Euler in 1742. More precisely, Goldbach claimed that “every number greater than 2 is an aggregate of three prime numbers.” (In Goldbach’s day, the convention was to consider 1 a prime number, so his statement is equivalent to the modern version in which the convention is to not include 1 among the prime numbers.)
Goldbach’s conjecture was published in English mathematician Edward Waring’s Meditationes algebraicae (1770), which also contained Waring’s problem and what was later known as Vinogradov’s theorem. The latter, which states that every sufficiently large odd integer can be expressed as the sum of three primes, was proved in 1937 by the Russian mathematician Ivan Matveyevich Vinogradov. Further progress on Goldbach’s conjecture occurred in 1973, when the Chinese mathematician Chen Jing Run proved that every sufficiently large even number is the sum of a prime and a number with at most two prime factors.
William L. Hosch
Encyclopædia Britannica. (n.d.). Goldbach conjecture. Britannica School. Retrieved August 30, 2021, from https://school.eb.com/levels/high/article/Goldbach-conjecture/471456