“The problem of the direct determination of the primitive roots of a prime number is one of the 'cruces' of the Theory of Numbers. Euler, who first observed the peculiarity of these numbers, has yet left us no rigorous proof of their existence; though assuming their existence, he succeeded in accurately determining their number. The defect in his demonstration was first supplied by Gauss, who has also proposed an indirect method for finding a primitive root.”

—  Henry John Stephen Smith

Report on the Theory of Numbers (1859) Part I, p. 49.
The Collected Mathematical Papers of Henry John Stephen Smith (1894) Vol. 1

Adopted from Wikiquote. Last update Sept. 14, 2021. History

Help us to complete the source, original and additional information

Do you have more details about the quote "The problem of the direct determination of the primitive roots of a prime number is one of the 'cruces' of the Theory o…" by Henry John Stephen Smith?
Henry John Stephen Smith photo
Henry John Stephen Smith 6
mathematician 1826–1883

Related quotes

“The Hindus introduced negative numbers… The first known use is Brahmagupta about 628; he also states the rules for the four operations with negative numbers. Bhāskara points out that the square root of a positive number is twofold, positive and negative. He brings up the matter of the square root of a negative number but says that there is no square root because a negative number is not a square. No definitions, axioms, or theorems are given.
The Hindus did not unreservedly accept negative numbers. Even Bhāskara, while giving 50 and -5 as two solutions of a problem, says, "The second value is in this case not to be taken, for it is inadequate; people do not approve of negative solutions."”

Morris Kline (1908–1992) American mathematician

However, negative numbers gained acceptance slowly.
Source: Mathematical Thought from Ancient to Modern Times (1972), p. 185.

“One of the first algebraists to accept negative numbers was… placed negative numbers on a par with positive numbers and gave both roots of a quadratic equation, even when both were negative. Both Girard and Harriot used the minus sign for the operation of subtraction and for negative numbers.”

Morris Kline (1908–1992) American mathematician

Source: Mathematical Thought from Ancient to Modern Times (1972), pp. 252-253.

“He treated Root exactly as he treated prime numbers. For him, primes were the base on which all other natural numbers relied; and children were the foundation of everything worthwhile in the adult world”

Yōko Ogawa book The Housekeeper and the Professor

Source: The Housekeeper and the Professor

“There exists among the most primitive tribes of Australia and Africa a system of numeration which has neither 5, 10, nor 20 for base. It is a binary system, i. e., of base two. These savages have not yet reached finger counting. They have independent numbers for one and two, and composite numbers up to six. Beyond six everything is denoted by “heap.””

Tobias Dantzig (1884–1956) American mathematician

Number: The Language of Science (1930)

Georg Cantor photo

“If there is some determinate succession of defined whole real numbers, among which there exists no greatest, on the basis of this second principle of generation a new number is obtained which is regarded as the limit of those numbers, i. e. is defined as the next greater number than all of them.”

Georg Cantor (1845–1918) mathematician, inventor of set theory

From Kant to Hilbert (1996)

Prasanta Chandra Mahalanobis photo

“We believe that the idea underlying this integral concept of statistics finds adequate expression in the ancient Indian work Sankhya in Sanskrit the usual meaning is ‘number‘, but the original root meaning was ‘determinate knowledge’ in the Atharva Veda a derivative from Sankhyata occurs both in the sense of ‘well-known‘ as well as ‘numbered’. The lexicons give both meanings. Amarakosa gives Sankhya – vicarana (deliberation, analysis) as well as ‘number’; also Sankhyavan”

Prasanta Chandra Mahalanobis (1893–1972) Indian scientist

panditah (wise, learned).
Quote, Prasanta Chandra Mahalanobis in Vigyanprasar

“You can ask the question about these ancient topics, such as perfect numbers and amicable numbers… and ask, are these good problems… I'd like to give a small amount of evidence… that they are… [S]tudying them helped us develop all of elementary number theory and from elementary number theory we developed the rest of number theory, and also you can argue that from elementary number theory came algebra..”

Carl Pomerance (1944) American mathematician

"Paul Erdős and the Rise of Statistical Thinking in Elementary Number Theory" https://www.youtube.com/watch?v=5cU0g9dI1S8&t=9m40s (July, 2013) Erdős Centennial Conference, Budapest.

James Whitbread Lee Glaisher photo

“Many of the greatest masters of the mathematical sciences were first attracted to mathematical inquiry by problems relating to numbers, and no one can glance at the periodicals of the present day which contain questions for solution without noticing how singular a charm such problems still continue to exert. The interest in numbers seems implanted in the human mind, and it is a pity that it should not have freer scope in this country. The methods of the theory of numbers 271 are peculiar to itself, and are not readily acquired by a student whose mind has for years been familiarized with the very different treatment which is appropriate to the theory of continuous magnitude; it is therefore extremely desirable that some portion of the theory should be included in the ordinary course of mathematical instruction at our University. From the moment that Gauss, in his wonderful treatise of 1801, laid down the true lines of the theory, it entered upon a new day, and no one is likely to be able to do useful work in any part of the subject who is unacquainted with the principles and conceptions with which he endowed it.”

James Whitbread Lee Glaisher (1848–1928) English mathematician and astronomer

Source: "Presidential Address British Association for the Advancement of Science," 1890, p. 467 : On the theory of numbers

Lucio Russo photo

“Euclid … manages to obtain a rigorous proof without ever dealing with infinity, by reducing the problem [of the infinitude of primes] to the study of finite numbers. This is exactly what contemporary mathematical analysis does.”

Lucio Russo (1944) Italian historian and scientist

2.4, "Discrete Mathematics and the Notion of Infinity", p. 45
The Forgotten Revolution: How Science Was Born in 300 BC and Why It Had to Be Reborn (2004)

Henry John Stephen Smith photo

“Legendre's Law of Quadratic Reciprocity' … is …the most important general truth in the science of integral numbers which has been discovered since the time of Fermat. It has been called by Gauss 'the gem of the higher arithmetic,' and is equally remarkable whether we consider the simplicity of its enunciation, the difficulties which for a long time attended its demonstration, or the number and variety of the results which have been obtained by its means. …[W]e find in the 'Opuscula Analytica' of Euler… a memoir… which contains a general and very elegant theorem from which the Law of Reciprocity is immediately deducible, and which is, vice versâ, deducible from that law. But Euler… expressly observes that the theorem is undemonstrated; and this would seem to be the only place in which he mentions it in connexion with the theory of the Residues of Powers; though in other researches he has frequently developed results which are consequences of the theorem, and which relate to the linear forms of the divisors of quadratic formulae. But here also his conclusions repose on induction only; though in one memoir he seems to have imagined… that he had obtained a satisfactory demonstration.”

Henry John Stephen Smith (1826–1883) mathematician

Report on the Theory of Numbers (1859) Part I, pp. 56-57.
The Collected Mathematical Papers of Henry John Stephen Smith (1894) Vol. 1

Related topics