The book is divided into seven sections, which are: Congruent Numbers in General Congruences of the First Degree Residues of Powers Congruences of the Second Degree Forms and Indeterminate Equations of the Second Degree Various Applications of the Preceding Discussions Equations Defining Sections of a Circle These sections are subdivided into numbered items, which sometimes state a theorem with proof, or otherwise develop a remark or thought. Although few of the results in these first sections are original, Gauss was the first mathematician to bring this material together and treat it in a systematic way. He also realized the importance of the property of unique factorization assured by the fundamental theorem of arithmetic , first studied by Euclid , which he restates and proves using modern tools. From Section IV onwards, much of the work is original.

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Gauss later solved this puzzle about his birthdate in the context of finding the date of Easter , deriving methods to compute the date in both past and future years. 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 students.

He completed his magnum opus , Disquisitiones Arithmeticae , in , at the age of 21—though it was not published until Gauss was so pleased with this result that he requested that a regular heptadecagon be inscribed on his tombstone.

The stonemason declined, stating that the difficult construction would essentially look like a circle. He discovered a construction of the heptadecagon on 30 March. On 8 April he became the first to prove the quadratic reciprocity law. This remarkably general law allows mathematicians to determine the solvability of any quadratic equation in modular arithmetic. The prime number theorem , conjectured on 31 May, gives a good understanding of how the prime numbers are distributed among the integers.

On 1 October he published a result on the number of solutions of polynomials with coefficients in finite fields , which years later led to the Weil conjectures. Highly developed convolutions were also found, which in the early 20th century were suggested as the explanation of his genius. In the days of his full strength, it furnished him recreation and, by the prospects which it opened up to him, gave consolation.

Toward the end of his life, it brought him confidence. To man is not vouchsafed that fullness of knowledge which would warrant his arrogantly holding that his blurred vision is the full light and that there can be none other which might report the truth as does his. For Gauss, not he who mumbles his creed, but he who lives it, is accepted. He believed that a life worthily spent here on earth is the best, the only, preparation for heaven.

Religion is not a question of literature, but of life. A book is inspired when it inspires. The unshakeable idea of personal continuance after death, the firm belief in a last regulator of things, in an eternal, just, omniscient, omnipotent God, formed the basis of his religious life, which harmonized completely with his scientific research.

In this work, Whewell had discarded the possibility of existing life in other planets, on the basis of theological arguments, but this was a position with which both Wagner and Gauss disagreed. Later Wagner explained that he did not fully believe in the Bible, though he confessed that he "envied" those who were able to easily believe.

He conceived spiritual life in the whole universe as a great system of law penetrated by eternal truth, and from this source he gained the firm confidence that death does not end all. He then married Minna Waldeck — [41] [42] on 4 August , [41] and had three more children. With Johanna — , his children were Joseph — , Wilhelmina — and Louis — With Minna Waldeck he also had three children: Eugene — , Wilhelm — and Therese — His mother lived in his house from until her death in He did not want any of his sons to enter mathematics or science for "fear of lowering the family name", as he believed none of them would surpass his own achievements.

They had an argument over a party Eugene held, which Gauss refused to pay for. The son left in anger and, in about , emigrated to the United States. Later, he moved to Missouri and became a successful businessman. Wilhelm also moved to America in and settled in Missouri, starting as a farmer and later becoming wealthy in the shoe business in St. Personality[ edit ] Gauss was an ardent perfectionist and a hard worker. He was never a prolific writer, refusing to publish work which he did not consider complete and above criticism.

This was in keeping with his personal motto pauca sed matura "few, but ripe". His personal diaries indicate that he had made several important mathematical discoveries years or decades before his contemporaries published them. Scottish-American mathematician and writer Eric Temple Bell said that if Gauss had published all of his discoveries in a timely manner, he would have advanced mathematics by fifty years. It is said that he attended only a single scientific conference, which was in Berlin in However, several of his students became influential mathematicians, among them Richard Dedekind and Bernhard Riemann.

Gauss supported the monarchy and opposed Napoleon , whom he saw as an outgrowth of revolution. Gauss summarized his views on the pursuit of knowledge in a letter to Farkas Bolyai dated 2 September as follows: It is not knowledge, but the act of learning, not possession but the act of getting there, which grants the greatest enjoyment. When I have clarified and exhausted a subject, then I turn away from it, in order to go into darkness again.

The never-satisfied man is so strange; if he has completed a structure, then it is not in order to dwell in it peacefully, but in order to begin another. I imagine the world conqueror must feel thus, who, after one kingdom is scarcely conquered, stretches out his arms for others.

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## Carl Friedrich Gauss

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## Disquisitiones Arithmeticae

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