Abstract
A prime number is a whole number (an integer) with the property that its only divisors are 1 and itself. By custom we do not consider 1 to be a prime.
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- 1.
More precisely, we need to divide by \(2, 3, 4,\ldots,\lfloor \sqrt{n}\rfloor \), where ⌊x⌋ denotes the floor function (also called the greatest integer function) equal to the largest integer less than or equal to x. We prefer to keep the notation cleaner by suppressing the floor function.
- 2.
The mass of the sun is about 1. 99 × 1030 kilograms. The mass of a proton or neutron is about 1. 67 × 10−27 kilograms. Thus there are about 1. 2 × 1057 protons and neutrons in the sun.
- 3.
In fact the Riemann hypothesis, first formulated by B. Riemann in 1859, is probably the most important unsolved problem in pure (abstract) mathematics. In its simplest form, the Riemann hypothesis (affectionately known to experts as RH) asserts that the prime numbers are randomly distributed. The more technical forms of the hypothesis are very concrete and technical estimates about the distribution of primes.
References and Further Reading
Agrawal, M., Kayal, N., Saxena, N.: PRIMES is in P. Annals of Mathematics 160, 781–793 (2004)
Bornemann, F.: PRIMES is in P: a breakthrough for everyman. Notices of the American Mathematical Society, vol. 50, pp. 545–552. (2003)
Carmichael, R.D.: On composite numbers p which satisfy the Fermat congruence a p−1 ≡ 1 m o d p. American Mathematical Monthly 19, 22–27 (1912)
Lenstra, Jr. H.W., Pomerance, C.: Primality testing with Gaussian periods. www.math.dartmouth.edu/~carlp/PDF/complexity12.pdf. Accessed July (2005)
Miller, G.L.: Riemann’s hypothesis and a test for primality. Journal of Computer and System Sciences 13, 300–317 (1976)
Rabin, M.O.: Probabilistic algorithm for testing primality. Journal of Number Theory 12, 128–138 (1980)
Solovay, R.M., Strassen, V.: A fast Monte-Carlo test for primality. SIAM Journal on Computing 6, 84–85 (1977)
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Krantz, S.G., Parks, H.R. (2014). Primality Testing. In: A Mathematical Odyssey. Springer, Boston, MA. https://doi.org/10.1007/978-1-4614-8939-9_11
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