# Lawrence Lugin, 1st Marquis of Krada

The Marquis of Krada | |
---|---|

Portrait of Lugin in 1646, hanging in the National Art Gallery in Samistopol | |

Born | |

Died | December 30, 1676 | (aged 66)

Education | University of Samistopol (A.H., 1630) |

Known for | Mathematics and physics: |

Scientific career | |

Fields | Mathematics, physics and philosophy |

Influences | Acacius, Theocritus, Xenagoras |

**Lawrence Lugin, 1st Marquis of Krada**, real name *Lavrentiy Avhust Lugin*, June 15, 1610 – December 30, 1676; aged 66) was a Narozalic physicist, mathematician, philosopher, scientist and theorist. A gifted child prodigy evident from a young age, his mother and father funded mathematical tutors to teach Lugin from a young age. Often regarded as the greatest mathematician and theorist of all time, Lugin's repertoire includes the discovery of universal gravitation, which revolutionised physics and how scientists and philosophers viewed the world, the Luginian triangle, a formulaic way of expanding binomials, and his outstanding contributions to number theory, algebra, pure mathematics, mechanics and analytic geometry. In philosophy, he is credited as one of the main philosophers of the neostoicist movement, but also dealt with topics such as ethics, metaphysics, creationism and teleology.

## Contents

## Early life and education

Lugin was born on June 15, 1610, to a Narozalic mother and father, both of whom were middle class, well-off people in Luchintsy, a port city in eastern Narozalica. He only spent a year of his life there, however, before he, his parents and his two sisters (Maria and Jasmine) relocated to the Narozalic capital of Samistopol on likely work related reasons for his father. Lugin's mother passed when he was six, shortly after his tutors had begun to teach the young Lugin. His tutors quickly remarked that Lugin had developed a remarkable aptitude for mathematics and sciences, particularly physics and mechanics, and displayed great understanding even at an early age. He continued to be taught by his tutors until the age of sixteen, where he applied for the University of Samistopol for a degree in applied mathematics.

While at the university, Lugin began documenting his first theorems and theories, in 1628, he documented his first written theorem, now known as Lugin's First Integer Theorem, which stated that no three positive integers a, b and c could satisfy the equation a^{n}+b^{n}=c^{n}, given n was a positive integer greater than two. His professor, amazed by the young Lugin's forward-thinking theorem, invited him to speak at a mathematics conference in Samistopol attended by the city's greatest mathematical minds and thinkers. Lugin presented his theorem at the conference and its attempts by various minds to solve the theorem at the conference opened up new routes of mathematical development. Lugin left the University of Samistopol in 1630, passing with flying colours and obtaining his degree in applied mathematics.

## Contributions to mathematics

Starting at university, and continuing through his life, Lugin had a profound and significant influence on mathematics and how it was observed, proposing many theorems and theories that progress mathematical thinking as a whole.

### Lugin' First Integer Theorem

Lugin first conjectured his First Integer Theorem in 1628, stating that no three positive integers *a*, *b* and *c* could satisfy the equation *a*^{n} + *b*^{n} = *c*^{n} where the number *n* is a postive integer greater than two. Lugin was aware that *n* = 1 and *n* = 2 had an infinite number of solutions that satisfied the equation.

The First Integer Theorem went unproved until Gaullican mathematician Jean-Jacques Delaplace released his first successful proof in 1997, which led to the further proving of the modularity theorem as well as large advances in mathematical morphisms, graphing and modularity lifting. The problem's inability to be solved also led to large advances throughout the mathematical world, with the development of the algebraic number theory being largely due to and stimulated by Lugin's First Integer Theorem and the theorem itself is one of the most famous in the mathematical world.

### Lugin' Second Integer Theorem

Lugin's Second Integer Theorem was developed while Lugin was vacationing on the banks of the Tsyr in the town of Litovizh, and states that if any integer *y* is a prime number, then for any integer *x*, the number *x ^{y}* -

*x*is an integer multiple of

*y*, expressed symbolically as:

*x*≡^{y}*x*(mod*y*).

However, when *x* is not divisible by *y*, Lugin's Second Integer Theorem is written as (as well as being equivalent to) the statement that *x*^{y - 1} - 1 is an integer multiple of *p*, symbolically expressed as:

*x*^{y - 1}≡ 1 (mod*y*).

Lugin's Second Integer Theorem was one of the first great contributions to the mathematical area of probability, contributing significantly to research within the field as well as being a fundamental result of elementary number theory. Lugin's Second Integer Theorem also led to the discovery of Luginian pseudoprimes, a class of numbers that share similar characteristics to prime numbers without actually being a prime number. Pseudoprimes are one of the results of unusual cases of Lugin's Second Integer Theorem.

### Luginian triangle

The Luginian triangle was developed in 1640 whilst Lugin was lecturing at the University of Solaria, particularly in the field of applied mathematics. It is a triangular array of binomial coefficients that descend such that a triangle with *a* on the top row and *b* and *c* on the bottom row would result in the guaranteed equation:

*b*+*c*≡*a*

The triangle continues indefinitely, and was the sole contributing factor to the later establishment of the binomial theorem, used for binomial expansions of brackets. The theorem combines the Luginian triangle with combination functions, such that:

- (
*x*+*y*)^{2}

would expand to give:

**1***x*^{2}*y*^{0}+**2***x*^{1}*y*^{1}+**1***x*^{0}*y*^{2}

This adheres to the second row of the Luginian triangle (the row of the triangle to use is denoted by the power the bracket is raised to), which progresses as 1 2 1, this works for any row of the triangle, and is particularly useful for expanding larger brackets, where pure expansion would be tedious and require the use of large numbers.

## Philosophy

As well as being a gifted mathematician, Lugin was also an avid philosopher, publishing multiple works on dealing with philosophical topics such as the self, existence and intrinsic value. One of the first modern philosophers, Lugin's works, treatises and books dealt extensively with the changing philosophical outlook in Euclea, mainly including topics such as ethics, metaphysics, creationism and teleology. Lugin was also one of the main philosophers behind the neostoicism movement in the mid-17th century, publishing his philosophical *magnum opus* *Partes eae fuerunt virtus* in 1650, considered one of the pillars of stoicist revivalism.

### Influences

Like most modern northern philosophers, Lugin drew significant influence from the triumvirate of Ancient Piraean philosophers, Acacius, Theocritus and Xenagoras. He also discussed the works of Solarian stoicist Ovidus in his works, and admired him as one of his favourite philosophers. Lugin often travelled to Tengaria, specifically the old site of Lasithi, a Piraean city located on the Tengarian coast, to devise his philosophical thoughts and often to concentrate on his mathematical works.

In *Partes eae fuerunt virtus*, Lugin points to Ovidus' works of stoic fiction, particularly *Medea* and *Thyestes*, which deal with themes of main characters repenting for mankind's sins. In line with other stoic philosophers, Lugin believed that virtue was the only real intrisic value within a person, and that other forms of happiness, mainly wealth and pleasure, are extrinsic and have no value in and of themselves. Lugin deemed them as *desideravit, sed necesse* (desired, but unnecessary) tenets of a person's happiness, regarding them as temporary and unfulfilling.

Lugin also studied the fundamental nature of reality and how it operates, part of general metaphysical philosophy but expanded upon with the advent of modern philosophy. Lugin believed in the creationist theory that the world was the result of divine acts and will, but believed that the fundamental operation of the world itself was independent of God. With his discovery and proof of universal gravitation in 1643, Lugin tied this in with the notion that God created the world to operate on its own, thereby disagreeing with the philosophical notion of the Will of God, putting him at odds with many other influential philosophers of the time.