“The obvious mathematical breakthrough would be development of an easy way to factor large prime numbers.”

Source: The Road Ahead (1995), p. 265 in hardcover edition, corrected in paperback

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American business magnate and philanthropist 1955

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“I would teach the world how the Greeks proved, more than 2,000 years ago, that there are infinitely many prime numbers. In my mind, this discovery is the beginning of mathematics – when humankind realised that, by pure thought alone, it could prove eternal truths of the universe.
Prime numbers are the indivisible numbers, numbers that can be divided only by themselves and one. They are the most important numbers in mathematics, because every number is built by multiplying prime numbers together – for example, 60 = 2 x 2 x 3 x 5. They are like the atoms of arithmetic, the hydrogen and oxygen of the world of numbers.”

Marcus du Sautoy (1965) British professor of mathematics

In "Life lessons" http://www.theguardian.com/science/2005/apr/07/science.highereducation?fb_ref=desktop The Guardian (7 April 2005)

“Factor analysis, i. e., isolation by way of mathematical analysis, of factors in multivariable phenomena in psychology and other fields”

Ludwig von Bertalanffy (1901–1972) austrian biologist and philosopher

General System Theory (1968), 4. Advances in General Systems Theory

“The problem of distinguishing prime numbers from composite numbers and of resolving the latter into their prime factors is known to be one of the most important and useful in arithmetic. It has engaged the industry and wisdom of ancient and modern geometers to such an extent that it would be superfluous to discuss the problem at length. … Further, the dignity of the science itself seems to require that every possible means be explored for the solution of a problem so elegant and so celebrated.”

Carl Friedrich Gauss book Disquisitiones Arithmeticae

Problema, numeros primos a compositis dignoscendi, hosque in factores suos primos resolvendi, ad gravissima ac utilissima totius arithmeticae pertinere, et geometrarum tum veterum tum recentiorum industriam ac sagacitatem occupavisse, tam notum est, ut de hac re copiose loqui superfluum foret. … [P]raetereaque scientiae dignitas requirere videtur, ut omnia subsidia ad solutionem problematis tam elegantis ac celebris sedulo excolantur.
Disquisitiones Arithmeticae (1801): Article 329

“Wherever mathematics has entered it has never again been pushed out by other developments. The mathematization of an area of human endeavor is not a passing fad; it is the prime mover of scientific and technological progress.”

Oskar Morgenstern (1902–1977) austrian economist

Oskar Morgenstern (Mathematica/Mathematic Policy Research), (from "A Look Back at Some of Our Contributions Over Time")

“In many different fields, empirical phenomena appear to obey a certain general law, which can be called the Law of Large Numbers. This law states that the ratios of numbers derived from the observation of a very large number of similar events remain practically constant, provided that these events are governed partly by constant factors and partly by variable factors whose variations are irregular and do not cause a systematic change in a definite direction.”

Siméon Denis Poisson (1781–1840) French mathematician, mechanician and physicist

Statement of Poisson's law also known as the Law of Large Numbers (1837), as quoted by [Richard Von Mises, Probability, Statistics and Truth, Allen and Unwin, 1957, 104-105]

“The distinction between micro- and macroinventions is useful because, as historians of technology emphasize, the word first is hazardous in this literature.. Many technological breakthroughs had a history that began before the event generally regarded as “the invention,” and almost all macroinventions required subsequent improvements to make them operational. Yet in a large number of cases, one or two identifiable events were crucial. Without such breakthroughs technological progress would eventually fizzle out.”

Joel Mokyr (1946) Israeli American economic historian

Source: The lever of riches: Technological creativity and economic progress, 1992, p. 14

“A large part of mathematics which becomes useful developed with absolutely no desire to be useful, and in a situation where nobody could possibly know in what area it would become useful; and there were no general indications that it ever would be so.”

John Von Neumann (1903–1957) Hungarian-American mathematician and polymath

"The Role of Mathematics in the Sciences and in Society" (1954) an address to Princeton alumni, published in John von Neumann : Collected Works (1963) edited by A. H. Taub ; also quoted in Out of the Mouths of Mathematicians : A Quotation Book for Philomaths (1993) by R. Schmalz
Context: A large part of mathematics which becomes useful developed with absolutely no desire to be useful, and in a situation where nobody could possibly know in what area it would become useful; and there were no general indications that it ever would be so. By and large it is uniformly true in mathematics that there is a time lapse between a mathematical discovery and the moment when it is useful; and that this lapse of time can be anything from 30 to 100 years, in some cases even more; and that the whole system seems to function without any direction, without any reference to usefulness, and without any desire to do things which are useful.

“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)

“Mathematics may be a way of developing physically, that is anatomically, new connections in the brain.”

Stanislaw Ulam (1909–1984) Polish-American mathematician

Source: Adventures of a Mathematician - Third Edition (1991), Chapter 15, Random Reflections on Mathematics and Science, p. 277

“I liked quantum mechanics very much. The subject was hard to understand but easy to apply to a large number of interesting problems.”

Willis Lamb (1913–2008) American Physicist

W. E. Lamb, Super classical quantum mechanics: the best interpretation of non relativistic quantum mechanics, Am. J. Phys. 69, 413-422 (2001).

“The obvious mathematical breakthrough would be development of an easy way to factor large prime numbers.”

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