78,557
78,557 (seventy-eight thousand five hundred [and] fifty-seven) is the natural number following 78,556 and preceding 78,558.[1]
In mathematics
Sierpiński number
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Unsolved problem in mathematics: Is 78,557 the smallest Sierpiński number? (more unsolved problems in mathematics)
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78557 is conjectured as the first, and smallest Sierpiński number.[2][3][4] It was proved to be one by John Selfridge in the year 1962, who showed that all numbers of the form 78557⋅2n + 1 have a factor in the covering set {3, 5, 7, 13, 19, 37, 73}.[5][6][2] But there are debates on whether 78557 is actually the smallest Sierpiński number. This is known as the Sierpiński problem.
To show that 78,557 really is the smallest Sierpiński number, one must show that all the odd numbers smaller than 78,557 are not Sierpiński numbers. That is, for every odd k below 78,557, there needs to exist a positive integer n such that k2n + 1 is prime.[6] The distributed volunteer computing project PrimeGrid is attempting to eliminate all the remaining values of k:[7]
- k = 21181, 22699, 24737, 55459, and 67607.[lower-alpha 1]
Notes
- ↑ See the current status for the remaining multipliers at PrimeGrid's website.
References
- ↑ Vanovschi, Vitalii. "Properties of the number 78557" (in en). http://www.numberempire.com:80/78557.
- ↑ 2.0 2.1 Koninck, Jean-Marie De (2009) (in en). Those Fascinating Numbers. American Mathematical Soc.. ISBN 978-0-8218-4807-4. https://books.google.com/books?hl=en&lr=&id=xdfTAwAAQBAJ&oi=fnd&pg=PR9&dq=Those+fascinating+numbers&ots=fUh1NubdD2&sig=lGyFqEb6EkVrv97nOS4sTp3g-Sw#v=onepage&q=78557&f=false.
- ↑ Sloane, N. J. A., ed. "Sequence A076336 ((Provable) Sierpiński numbers: odd numbers n such that for all k >= 1 the numbers n*2^k + 1 are composite)". OEIS Foundation. https://oeis.org/A076336.
- ↑ Weisstein, Eric W.. "Sierpiński's Composite Number Theorem" (in en). Wolfram Research, Inc.. https://mathworld.wolfram.com/SierpinskisCompositeNumberTheorem.html.
- ↑ Filaseta, Michael; Finch, Carrie; Kozek, Mark (July 2008). "On powers associated with Sierpiński numbers, Riesel numbers and Polignac's conjecture" (in en). Journal of Number Theory 128 (7): 1916–1940. doi:10.1016/j.jnt.2008.02.004. https://linkinghub.elsevier.com/retrieve/pii/S0022314X08000462.
- ↑ 6.0 6.1 "The Prime Glossary: Sierpinski number". https://t5k.org/glossary/page.php?sort=SierpinskiNumber.
- ↑ "Seventeen or Bust statistics". https://www.primegrid.com/stats_sob_llr.php.
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