# Carl Friedrich Gauss

**GaussCarl GaussCarl Friedrich GaußC. F. GaussKarl Friedrich GaussJohann Carl Friedrich GaussC.F. GaussGauss, Carl FriedrichCarl F. GaussGaussian**

Johann Carl Friedrich Gauss (Gauß ; Carolus Fridericus Gauss; 30 April 1777 – 23 February 1855) was a German mathematician and physicist who made significant contributions to many fields in mathematics and sciences.wikipedia

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### Disquisitiones Arithmeticae

He completed his magnum opus, Disquisitiones Arithmeticae, in 1798, at the age of 21—though it was not published until 1801.

The Disquisitiones Arithmeticae (Latin for "Arithmetical Investigations") is a textbook of number theory written in Latin by Carl Friedrich Gauss in 1798 when Gauss was 21 and first published in 1801 when he was 24.

### List of German mathematicians

**German mathematician**

Johann Carl Friedrich Gauss (Gauß ; Carolus Fridericus Gauss; 30 April 1777 – 23 February 1855) was a German mathematician and physicist who made significant contributions to many fields in mathematics and sciences.

### Modular arithmetic

**modulomodcongruent**

He further advanced modular arithmetic, greatly simplifying manipulations in number theory. for congruence and used it in a clean presentation of modular arithmetic, contained the first two proofs of the law of quadratic reciprocity, developed the theories of binary and ternary quadratic forms, stated the class number problem for them, and showed that a regular heptadecagon (17-sided polygon) can be constructed with straightedge and compass.

The modern approach to modular arithmetic was developed by Carl Friedrich Gauss in his book Disquisitiones Arithmeticae, published in 1801.

### Straightedge and compass construction

**compass and straightedgecompass and straightedge constructionscompass-and-straightedge construction**

His breakthrough occurred in 1796 when he showed that a regular polygon can be constructed by compass and straightedge if the number of its sides is the product of distinct Fermat primes and a power of 2. for congruence and used it in a clean presentation of modular arithmetic, contained the first two proofs of the law of quadratic reciprocity, developed the theories of binary and ternary quadratic forms, stated the class number problem for them, and showed that a regular heptadecagon (17-sided polygon) can be constructed with straightedge and compass.

Gauss showed that some polygons are constructible but that most are not.

### Technical University of Braunschweig

**Braunschweig University of TechnologyTU BraunschweigCollegium Carolinum**

Gauss's intellectual abilities attracted the attention of the Duke of Brunswick, who sent him to the Collegium Carolinum (now Braunschweig University of Technology), which he attended from 1792 to 1795, and to the University of Göttingen from 1795 to 1798.

Current and former members of the TU Braunschweig include the mathematicians Carl Friedrich Gauss and Richard Dedekind, Nobel laureates Klaus von Klitzing, Manfred Eigen, and Georg Wittig, aerospace engineer Adolf Busemann, the former CEOs of SAP, Metro Group, and Henkel (Henning Kagermann, Erwin Conradi, and Konrad Henkel), BMW CEO Harald Krüger, Porsche CEO Oliver Blume, Airbus Defence and Space CEO Dirk Hoke, Siemens Mobility CEO Michael Peter, former Eurocopter CEO Lutz Bertling, and truck engineer and entrepreneur Heinrich Büssing of Büssing AG, engineer and founder of Claas KGaA mbH August Claas and world-renowned architect Meinhard von Gerkan.

### Quadratic reciprocity

**law of quadratic reciprocityquadratic reciprocity lawAureum Theorema**

On 8 April he became the first to prove the quadratic reciprocity law. for congruence and used it in a clean presentation of modular arithmetic, contained the first two proofs of the law of quadratic reciprocity, developed the theories of binary and ternary quadratic forms, stated the class number problem for them, and showed that a regular heptadecagon (17-sided polygon) can be constructed with straightedge and compass.

The theorem was conjectured by Euler and Legendre and first proved by Gauss.

### Triangular number

**triangular numberstriangulartriangle number**

Gauss also discovered that every positive integer is representable as a sum of at most three triangular numbers on 10 July and then jotted down in his diary the note: "ΕΥΡΗΚΑ! . On 1 October he published a result on the number of solutions of polynomials with coefficients in finite fields, which 150 years later led to the Weil conjectures.

Carl Friedrich Gauss is said to have found this relationship in his early youth, by multiplying

### Gauss's diary

**his diaryGauss' diary**

Gauss also discovered that every positive integer is representable as a sum of at most three triangular numbers on 10 July and then jotted down in his diary the note: "ΕΥΡΗΚΑ! . On 1 October he published a result on the number of solutions of polynomials with coefficients in finite fields, which 150 years later led to the Weil conjectures.

Gauss's diary was a record of the mathematical discoveries of German mathematician Carl Friedrich Gauss from 1796 to 1814.

### Prime number theorem

**distribution of primesdistribution of prime numbersprime number theorem for arithmetic progressions**

The prime number theorem, conjectured on 31 May, gives a good understanding of how the prime numbers are distributed among the integers.

. Carl Friedrich Gauss considered the same question at age 15 or 16 "in the year 1792 or 1793", according to his own recollection in 1849.

### Weil conjectures

**Riemann hypothesis for zeta-functionsWeil conjecture**

Gauss also discovered that every positive integer is representable as a sum of at most three triangular numbers on 10 July and then jotted down in his diary the note: "ΕΥΡΗΚΑ! . On 1 October he published a result on the number of solutions of polynomials with coefficients in finite fields, which 150 years later led to the Weil conjectures.

The earliest antecedent of the Weil conjectures is by Carl Friedrich Gauss and appears in section VII of his Disquisitiones Arithmeticae, concerned with roots of unity and Gaussian periods.

### Arithmetic progression

**arithmetic sequencearithmetic seriesarithmetic**

In his memorial on Gauss, Wolfgang Sartorius von Waltershausen says that when Gauss was barely three years old he corrected a math error his father made; and that when he was seven, he confidently solved an arithmetic series problem faster than anyone else in his class of 100 students.

According to an anecdote, young Carl Friedrich Gauss reinvented this method to compute the sum 1+2+3+...+99+100 for a punishment in primary school.

### Heptadecagon

**17-gon17regular 17-gon**

Gauss was so pleased with this result that he requested that a regular heptadecagon be inscribed on his tombstone. for congruence and used it in a clean presentation of modular arithmetic, contained the first two proofs of the law of quadratic reciprocity, developed the theories of binary and ternary quadratic forms, stated the class number problem for them, and showed that a regular heptadecagon (17-sided polygon) can be constructed with straightedge and compass.

As 17 is a Fermat prime, the regular heptadecagon is a constructible polygon (that is, one that can be constructed using a compass and unmarked straightedge): this was shown by Carl Friedrich Gauss in 1796 at the age of 19.

### Charles William Ferdinand, Duke of Brunswick

**Duke of BrunswickCharles William Ferdinand, Duke of Brunswick-WolfenbüttelCharles William Ferdinand**

Gauss's intellectual abilities attracted the attention of the Duke of Brunswick, who sent him to the Collegium Carolinum (now Braunschweig University of Technology), which he attended from 1792 to 1795, and to the University of Göttingen from 1795 to 1798.

He sponsored enlightenment arts and sciences; most notably he was patron to the young mathematician Carl Friedrich Gauss, paying for him to attend university against the wishes of Gauss' father.

### Bernhard Riemann

**RiemannGeorg Friedrich Bernhard RiemannRiemann, Bernhard**

In 1854, Gauss selected the topic for Bernhard Riemann's inaugural lecture "Über die Hypothesen, welche der Geometrie zu Grunde liegen" (About the hypotheses that underlie Geometry). However, several of his students became influential mathematicians, among them Richard Dedekind and Bernhard Riemann.

However, once there, he began studying mathematics under Carl Friedrich Gauss (specifically his lectures on the method of least squares).

### Sophie Germain

**GermainGermain, SophieHonors in number theory**

Before she died, Sophie Germain was recommended by Gauss to receive her honorary degree; she never received it.

Despite initial opposition from her parents and difficulties presented by society, she gained education from books in her father's library, including ones by Leonhard Euler, and from correspondence with famous mathematicians such as Lagrange, Legendre, and Gauss.

### Albanifriedhof

**Albani CemeteryAlbanifriedhof Cemetery**

On 23 February 1855, Gauss died of a heart attack in Göttingen (then Kingdom of Hanover and now Lower Saxony); he is interred in the Albani Cemetery there.

It is most famous as the final resting place of Carl Friedrich Gauss.The cemetery is named after St. Albani Evangelical Lutheran Church in Göttingen.

### Richard Dedekind

**DedekindJulius Wilhelm Richard DedekindR. Dedekind**

However, several of his students became influential mathematicians, among them Richard Dedekind and Bernhard Riemann.

Gauss was still teaching, although mostly at an elementary level, and Dedekind became his last student.

### Braunschweig

**BrunswickBrunswick, GermanyBraunschweig, Germany**

Johann Carl Friedrich Gauss was born on 30 April 1777 in Brunswick (Braunschweig), in the Duchy of Brunswick-Wolfenbüttel (now part of Lower Saxony, Germany), to poor, working-class parents.

It had such famous pupils as Carl Friedrich Gauss, Hoffmann von Fallersleben, Richard Dedekind and Louis Spohr.

### Class number problem

**class number 1 problemclass number problem for imaginary quadratic fieldsclass number 1**

for congruence and used it in a clean presentation of modular arithmetic, contained the first two proofs of the law of quadratic reciprocity, developed the theories of binary and ternary quadratic forms, stated the class number problem for them, and showed that a regular heptadecagon (17-sided polygon) can be constructed with straightedge and compass.

It is named after Carl Friedrich Gauss.

### Heinrich Ewald

**EwaldG.H.A. EwaldGeorg Heinrich August von Ewald**

Two people gave eulogies at his funeral: Gauss's son-in-law Heinrich Ewald, and Wolfgang Sartorius von Waltershausen, who was Gauss's close friend and biographer.

Heinrich Ewald married in 1830 Wilhelmina (1808–1846), daughter of C.F. Gauss.

### Friedrich Bessel

**Friedrich Wilhelm BesselBesselBessel, Friedrich**

On Gauss's recommendation, Friedrich Bessel was awarded an honorary doctor degree from Göttingen in March 1811.

On the recommendation of fellow mathematician and physicist Carl Friedrich Gauss (with whom he regularly corresponded) he was awarded an honorary doctor degree from the University of Göttingen in March 1811.

### G. Waldo Dunnington

**Dunnington, G. Waldo**

One of his biographers, G. Waldo Dunnington, described Gauss's religious views as follows:

Guy Waldo Dunnington (January 15, 1906, Bowling Green, Missouri – April 10, 1974, Natchitoches, Louisiana) was a writer, historian and professor of German known for his writings on the famous German mathematician Carl Friedrich Gauss.

### Fundamental theorem of algebra

**boundedFundamental- of Algebranot an algebraic concept**

In his 1799 doctorate in absentia, A new proof of the theorem that every integral rational algebraic function of one variable can be resolved into real factors of the first or second degree, Gauss proved the fundamental theorem of algebra which states that every non-constant single-variable polynomial with complex coefficients has at least one complex root.

The other one was published by Gauss in 1799 and it was mainly geometric, but it had a topological gap, filled by Alexander Ostrowski in 1920, as discussed in Smale (1981).

### Fermat number

**Fermat primesFermat numbersGeneralized Fermat prime**

His breakthrough occurred in 1796 when he showed that a regular polygon can be constructed by compass and straightedge if the number of its sides is the product of distinct Fermat primes and a power of 2.

Carl Friedrich Gauss developed the theory of Gaussian periods in his Disquisitiones Arithmeticae and formulated a sufficient condition for the constructibility of regular polygons.

### Cooley–Tukey FFT algorithm

**Cooley–TukeyCooley–Tukey algorithmfast Fourier transform**

The algorithm, along with its recursive application, was invented by Carl Friedrich Gauss.