Mukammal son (Uzbek Wikipedia)

Analysis of information sources in references of the Wikipedia article "Mukammal son" in Uzbek language version.

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10doi.org

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4archive.org

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4web.archive.org

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4harvard.edu

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2mersenne.org

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2colgate.edu

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2ams.org

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2arxiv.org

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2byu.edu

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1utm.edu

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1st-and.ac.uk

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1torreys.org

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1books.google.com

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1wolframalpha.com

4,660^th place

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1quantamagazine.org

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1austms.org.au

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1lirmm.fr

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1worldcat.org

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1zbmath.org

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1westga.edu

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1albany.edu

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1tus.ac.jp

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albany.edu

nyjm.albany.edu

Pomerance, Carl; Luca, Florian (2010). „On the radical of a perfect number“. New York Journal of Mathematics. 16-jild. 23–30-bet. Qaraldi: 7 December 2018.

ams.org

Iannucci, DE (1999). „The second largest prime divisor of an odd perfect number exceeds ten thousand“ (PDF). Mathematics of Computation. 68-jild, № 228. 1749–1760-bet. Bibcode:1999MaCom..68.1749I. doi:10.1090/S0025-5718-99-01126-6. Qaraldi: 30 March 2011.
Iannucci, DE (2000). „The third largest prime divisor of an odd perfect number exceeds one hundred“ (PDF). Mathematics of Computation. 69-jild, № 230. 867–879-bet. Bibcode:2000MaCom..69..867I. doi:10.1090/S0025-5718-99-01127-8. Qaraldi: 30 March 2011.

archive.org

Dickson, L. E.. History of the Theory of Numbers, Vol. I. Washington: Carnegie Institution of Washington, 1919 — 4 bet.
In Introduction to Arithmetic, Chapter 16, he says of perfect numbers, „There is a method of producing them, neat and unfailing, which neither passes by any of the perfect numbers nor fails to differentiate any of those that are not such, which is carried out in the following way.“ He then goes on to explain a procedure which is equivalent to finding a triangular number based on a Mersenne prime.
Dickson, L. E.. History of the Theory of Numbers, Vol. I. Washington: Carnegie Institution of Washington, 1919 — 10 bet.
Dickson, L. E.. History of the Theory of Numbers, Vol. I. Washington: Carnegie Institution of Washington, 1919 — 6 bet.

arxiv.org

Zelinsky, Joshua (2019-yil iyul). „Upper bounds on the second largest prime factor of an odd perfect number“. International Journal of Number Theory. 15-jild, № 6. 1183–1189-bet. arXiv:1810.11734. doi:10.1142/S1793042119500659. {{cite magazine}}: sana kiritilishi kerak boʻlgan parametrga berilgan qiymatni tekshirish lozim: |date= (yordam).
Nielsen, Pace P. (2007). „Odd perfect numbers have at least nine distinct prime factors“ (PDF). Mathematics of Computation. 76-jild, № 260. 2109–2126-bet. arXiv:math/0602485. Bibcode:2007MaCom..76.2109N. doi:10.1090/S0025-5718-07-01990-4. 2021-11-03da asl nusxadan (PDF) arxivlandi. Qaraldi: 30 March 2011.

austms.org.au

Roberts, T (2008). „On the Form of an Odd Perfect Number“ (PDF). Australian Mathematical Gazette. 35-jild, № 4. 244-bet.

books.google.com

Pickover, C. Wonders of Numbers: Adventures in Mathematics, Mind, and Meaning. Oxford: Oxford University Press, 2001 — 360 bet. ISBN 0-19-515799-0.

byu.edu

math.byu.edu

Nielsen, Pace P. (2015). „Odd perfect numbers, Diophantine equations, and upper bounds“ (PDF). Mathematics of Computation. 84-jild, № 295. 2549–2567-bet. doi:10.1090/S0025-5718-2015-02941-X. 2015-07-08da asl nusxadan (PDF) arxivlandi. Qaraldi: 13 August 2015.
Nielsen, Pace P. (2007). „Odd perfect numbers have at least nine distinct prime factors“ (PDF). Mathematics of Computation. 76-jild, № 260. 2109–2126-bet. arXiv:math/0602485. Bibcode:2007MaCom..76.2109N. doi:10.1090/S0025-5718-07-01990-4. 2021-11-03da asl nusxadan (PDF) arxivlandi. Qaraldi: 30 March 2011.

colgate.edu

math.colgate.edu

Zelinsky, Joshua (3–avgust 2021–yil). „On the Total Number of Prime Factors of an Odd Perfect Number“ (PDF). Integers. 21-jild. Qaraldi: 7 August 2021.{{cite magazine}}: CS1 maint: date format ()
Bibby, Sean; Vyncke, Pieter; Zelinsky, Joshua (23–noyabr 2021–yil). „On the Third Largest Prime Divisor of an Odd Perfect Number“ (PDF). Integers. 21-jild. Qaraldi: 6 December 2021.{{cite magazine}}: CS1 maint: date format ()

doi.org

Ochem, Pascal; Rao, Michaël (2012). „Odd perfect numbers are greater than 10¹⁵⁰⁰“ (PDF). Mathematics of Computation. 81-jild, № 279. 1869–1877-bet. doi:10.1090/S0025-5718-2012-02563-4. ISSN 0025-5718. Zbl 1263.11005.
Chen, Yong-Gao; Tang, Cui-E (2014). „Improved upper bounds for odd multiperfect numbers“. Bulletin of the Australian Mathematical Society. 89-jild, № 3. 353–359-bet. doi:10.1017/S0004972713000488.
Ochem, Pascal; Rao, Michaël (2014). „On the number of prime factors of an odd perfect number“. Mathematics of Computation. 83-jild, № 289. 2435–2439-bet. doi:10.1090/S0025-5718-2013-02776-7.
Goto, T; Ohno, Y (2008). „Odd perfect numbers have a prime factor exceeding 10⁸“ (PDF). Mathematics of Computation. 77-jild, № 263. 1859–1868-bet. Bibcode:2008MaCom..77.1859G. doi:10.1090/S0025-5718-08-02050-9. 7–avgust 2011–yilda asl nusxadan (PDF) arxivlandi. Qaraldi: 30 March 2011.{{cite magazine}}: CS1 maint: date format ()
Konyagin, Sergei; Acquaah, Peter (2012). „On Prime Factors of Odd Perfect Numbers“. International Journal of Number Theory. 8-jild, № 6. 1537–1540-bet. doi:10.1142/S1793042112500935.
Iannucci, DE (1999). „The second largest prime divisor of an odd perfect number exceeds ten thousand“ (PDF). Mathematics of Computation. 68-jild, № 228. 1749–1760-bet. Bibcode:1999MaCom..68.1749I. doi:10.1090/S0025-5718-99-01126-6. Qaraldi: 30 March 2011.
Zelinsky, Joshua (2019-yil iyul). „Upper bounds on the second largest prime factor of an odd perfect number“. International Journal of Number Theory. 15-jild, № 6. 1183–1189-bet. arXiv:1810.11734. doi:10.1142/S1793042119500659. {{cite magazine}}: sana kiritilishi kerak boʻlgan parametrga berilgan qiymatni tekshirish lozim: |date= (yordam).
Iannucci, DE (2000). „The third largest prime divisor of an odd perfect number exceeds one hundred“ (PDF). Mathematics of Computation. 69-jild, № 230. 867–879-bet. Bibcode:2000MaCom..69..867I. doi:10.1090/S0025-5718-99-01127-8. Qaraldi: 30 March 2011.
Nielsen, Pace P. (2015). „Odd perfect numbers, Diophantine equations, and upper bounds“ (PDF). Mathematics of Computation. 84-jild, № 295. 2549–2567-bet. doi:10.1090/S0025-5718-2015-02941-X. 2015-07-08da asl nusxadan (PDF) arxivlandi. Qaraldi: 13 August 2015.
Nielsen, Pace P. (2007). „Odd perfect numbers have at least nine distinct prime factors“ (PDF). Mathematics of Computation. 76-jild, № 260. 2109–2126-bet. arXiv:math/0602485. Bibcode:2007MaCom..76.2109N. doi:10.1090/S0025-5718-07-01990-4. 2021-11-03da asl nusxadan (PDF) arxivlandi. Qaraldi: 30 March 2011.

harvard.edu

ui.adsabs.harvard.edu

Goto, T; Ohno, Y (2008). „Odd perfect numbers have a prime factor exceeding 10⁸“ (PDF). Mathematics of Computation. 77-jild, № 263. 1859–1868-bet. Bibcode:2008MaCom..77.1859G. doi:10.1090/S0025-5718-08-02050-9. 7–avgust 2011–yilda asl nusxadan (PDF) arxivlandi. Qaraldi: 30 March 2011.{{cite magazine}}: CS1 maint: date format ()
Iannucci, DE (1999). „The second largest prime divisor of an odd perfect number exceeds ten thousand“ (PDF). Mathematics of Computation. 68-jild, № 228. 1749–1760-bet. Bibcode:1999MaCom..68.1749I. doi:10.1090/S0025-5718-99-01126-6. Qaraldi: 30 March 2011.
Iannucci, DE (2000). „The third largest prime divisor of an odd perfect number exceeds one hundred“ (PDF). Mathematics of Computation. 69-jild, № 230. 867–879-bet. Bibcode:2000MaCom..69..867I. doi:10.1090/S0025-5718-99-01127-8. Qaraldi: 30 March 2011.
Nielsen, Pace P. (2007). „Odd perfect numbers have at least nine distinct prime factors“ (PDF). Mathematics of Computation. 76-jild, № 260. 2109–2126-bet. arXiv:math/0602485. Bibcode:2007MaCom..76.2109N. doi:10.1090/S0025-5718-07-01990-4. 2021-11-03da asl nusxadan (PDF) arxivlandi. Qaraldi: 30 March 2011.

lirmm.fr

Ochem, Pascal; Rao, Michaël (2012). „Odd perfect numbers are greater than 10¹⁵⁰⁰“ (PDF). Mathematics of Computation. 81-jild, № 279. 1869–1877-bet. doi:10.1090/S0025-5718-2012-02563-4. ISSN 0025-5718. Zbl 1263.11005.

mersenne.org

GIMPS Milestones Report. Retrieved 2018-02-27
„GIMPS Home“. Mersenne.org. Qaraldi: 21-iyul 2022-yil.

quantamagazine.org

Nadis, Steve. „Mathematicians Open a New Front on an Ancient Number Problem“. Quanta Magazine (10-sentabr 2020-yil). Qaraldi: 10-sentabr 2020-yil.

st-and.ac.uk

www-groups.dcs.st-and.ac.uk

„Perfect numbers“. www-groups.dcs.st-and.ac.uk. 2-noyabr 2017-yilda asl nusxadan arxivlangan. Qaraldi: 9-may 2018-yil.

torreys.org

Rogers, Justin M. (2015). "The Reception of Philonic Arithmological Exegesis in Didymus the Blind's Commentary on Genesis". http://torreys.org/sblpapers2015/S22-05_philonic_arithmological_exegesis.pdf.

tus.ac.jp

ma.noda.tus.ac.jp

Goto, T; Ohno, Y (2008). „Odd perfect numbers have a prime factor exceeding 10⁸“ (PDF). Mathematics of Computation. 77-jild, № 263. 1859–1868-bet. Bibcode:2008MaCom..77.1859G. doi:10.1090/S0025-5718-08-02050-9. 7–avgust 2011–yilda asl nusxadan (PDF) arxivlandi. Qaraldi: 30 March 2011.{{cite magazine}}: CS1 maint: date format ()

utm.edu

primes.utm.edu

Caldwell, Chris, „A proof that all even perfect numbers are a power of two times a Mersenne prime“.

web.archive.org

„Perfect numbers“. www-groups.dcs.st-and.ac.uk. 2-noyabr 2017-yilda asl nusxadan arxivlangan. Qaraldi: 9-may 2018-yil.
Goto, T; Ohno, Y (2008). „Odd perfect numbers have a prime factor exceeding 10⁸“ (PDF). Mathematics of Computation. 77-jild, № 263. 1859–1868-bet. Bibcode:2008MaCom..77.1859G. doi:10.1090/S0025-5718-08-02050-9. 7–avgust 2011–yilda asl nusxadan (PDF) arxivlandi. Qaraldi: 30 March 2011.{{cite magazine}}: CS1 maint: date format ()
Nielsen, Pace P. (2015). „Odd perfect numbers, Diophantine equations, and upper bounds“ (PDF). Mathematics of Computation. 84-jild, № 295. 2549–2567-bet. doi:10.1090/S0025-5718-2015-02941-X. 2015-07-08da asl nusxadan (PDF) arxivlandi. Qaraldi: 13 August 2015.
Nielsen, Pace P. (2007). „Odd perfect numbers have at least nine distinct prime factors“ (PDF). Mathematics of Computation. 76-jild, № 260. 2109–2126-bet. arXiv:math/0602485. Bibcode:2007MaCom..76.2109N. doi:10.1090/S0025-5718-07-01990-4. 2021-11-03da asl nusxadan (PDF) arxivlandi. Qaraldi: 30 March 2011.

westga.edu

Nielsen, Pace P. (2003). „An upper bound for odd perfect numbers“. Integers. 3-jild. A14–A22-bet. Qaraldi: 23 March 2021.

wolframalpha.com

„Number of primes <= 43112609“. Wolfram Alpha. Qaraldi: 28-oktabr 2018-yil.

worldcat.org

Ochem, Pascal; Rao, Michaël (2012). „Odd perfect numbers are greater than 10¹⁵⁰⁰“ (PDF). Mathematics of Computation. 81-jild, № 279. 1869–1877-bet. doi:10.1090/S0025-5718-2012-02563-4. ISSN 0025-5718. Zbl 1263.11005.

zbmath.org

Ochem, Pascal; Rao, Michaël (2012). „Odd perfect numbers are greater than 10¹⁵⁰⁰“ (PDF). Mathematics of Computation. 81-jild, № 279. 1869–1877-bet. doi:10.1090/S0025-5718-2012-02563-4. ISSN 0025-5718. Zbl 1263.11005.