تاريخ الرياضيات
الاعداد و نظريتها
تاريخ التحليل
تار يخ الجبر
الهندسة و التبلوجي
الرياضيات في الحضارات المختلفة
العربية
اليونانية
البابلية
الصينية
المايا
المصرية
الهندية
الرياضيات المتقطعة
المنطق
اسس الرياضيات
فلسفة الرياضيات
مواضيع عامة في المنطق
الجبر
الجبر الخطي
الجبر المجرد
الجبر البولياني
مواضيع عامة في الجبر
الضبابية
نظرية المجموعات
نظرية الزمر
نظرية الحلقات والحقول
نظرية الاعداد
نظرية الفئات
حساب المتجهات
المتتاليات-المتسلسلات
المصفوفات و نظريتها
المثلثات
الهندسة
الهندسة المستوية
الهندسة غير المستوية
مواضيع عامة في الهندسة
التفاضل و التكامل
المعادلات التفاضلية و التكاملية
معادلات تفاضلية
معادلات تكاملية
مواضيع عامة في المعادلات
التحليل
التحليل العددي
التحليل العقدي
التحليل الدالي
مواضيع عامة في التحليل
التحليل الحقيقي
التبلوجيا
نظرية الالعاب
الاحتمالات و الاحصاء
نظرية التحكم
بحوث العمليات
نظرية الكم
الشفرات
الرياضيات التطبيقية
نظريات ومبرهنات
علماء الرياضيات
500AD
500-1499
1000to1499
1500to1599
1600to1649
1650to1699
1700to1749
1750to1779
1780to1799
1800to1819
1820to1829
1830to1839
1840to1849
1850to1859
1860to1864
1865to1869
1870to1874
1875to1879
1880to1884
1885to1889
1890to1894
1895to1899
1900to1904
1905to1909
1910to1914
1915to1919
1920to1924
1925to1929
1930to1939
1940to the present
علماء الرياضيات
الرياضيات في العلوم الاخرى
بحوث و اطاريح جامعية
هل تعلم
طرائق التدريس
الرياضيات العامة
نظرية البيان
Repunit
المؤلف:
Ball, W. W. R. and Coxeter, H. S. M
المصدر:
Mathematical Recreations and Essays, 13th ed. New York: Dover
الجزء والصفحة:
...
27-9-2020
1395
+
-
20
Repunit
A repunit is a number consisting of copies of the single digit 1. The term "repunit" was coined by Beiler (1966), who also gave the first tabulation of known factors.
In base-10, repunits have the form
 |
 |
 |
(1) |
 |
 |
 |
(2) |
Repunits 
therefore have exactly 
decimal digits. Amazingly, the squares of the repunits 
give the Demlo numbers, 
, 
, 
, ... (OEIS A002275 and A002477).
The number of factors for the base-10 repunits for 
, 2, ... are 1, 1, 2, 2, 2, 5, 2, 4, 4, 4, 2, 7, 3, ... (OEIS A046053). The base-10 repunit probable primes 
occur for 
, 19, 23, 317, and 1031, 49081, 86453, 109297, and 270343 (OEIS A004023; Madachy 1979, Williams and Dubner 1986, Ball and Coxeter 1987, Granlund, Dubner 1999, Baxter 2000), where 
is the largest proven prime (Williams and Dubner 1986). T. Granlund completed a search up to 
in 1998 using two months of CPU time on a parallel computer. The search was extended by Dubner (1999), culminating in the discovery of the probable prime 
. A number of larger repunit probable primes have since been found, as summarized in the following table.
 |
discoverer(s) | date |
| 49081 | H. Dubner (1999, 2002) | Sep. 9, 1999 |
| 86453 | L. Baxter (2000) | Oct. 26, 2000 |
| 109297 | P. Bourdelais (2007), H. Dubner (2007) | Mar. 26-28, 2007 |
| 270343 | M. Voznyy and A. Budnyy (2007) | Jul. 11, 2007 |
Every prime repunit is a circular prime.
Repunit can be generalized to base 
, giving a base-
repunit as number of the form
 |
(3) |
This gives the special cases summarized in the following table.
 |
 |
name |
| 2 |  |
Mersenne number  |
| 10 |  |
repunit  |
The idea of repunits can also be extended to negative bases. Except for requiring 
to be odd, the math is very similar (Dubner and Granlund 2000).
 |
OEIS |  -repunits |
 |
A066443 | 1, 7, 61, 547, 4921, 44287, 398581, ... |
 |
A007583 | 1, 3, 11, 43, 171, 683, 2731, ... |
| 2 | A000225 | 1, 3, 7, 15, 31, 63, 127, ... |
| 3 | A003462 | 1, 4, 13, 40, 121, 364, ... |
| 4 | A002450 | 1, 5, 21, 85, 341, 1365, ... |
| 5 | A003463 | 1, 6, 31, 156, 781, 3906, ... |
| 6 | A003464 | 1, 7, 43, 259, 1555, 9331, ... |
| 7 | A023000 | 1, 8, 57, 400, 2801, 19608, ... |
| 8 | A023001 | 1, 9, 73, 585, 4681, 37449, ... |
| 9 | A002452 | 1, 10, 91, 820, 7381, 66430, ... |
| 10 | A002275 | 1, 11, 111, 1111, 11111, ... |
| 11 | A016123 | 1, 12, 133, 1464, 16105, 177156, ... |
| 12 | A016125 | 1, 13, 157, 1885, 22621, 271453, ... |
Williams and Seah (1979) factored generalized repunits for 
and 
. A (base-10) repunit can be prime only if 
is prime, since otherwise 
is a binomial number which can be factored algebraically. In fact, if 
is even, then 
. As with positive bases, all the exponents of prime repunits with negative bases are also prime.
 |
OEIS |  of prime  -repunits |
 |
A057178 | 5, 11, 109, 193, 1483, ... |
 |
A057177 | 5, 7, 179, 229, 439, 557, 6113, ... |
 |
A001562 | 5, 7, 19, 31, 53, 67, 293, ... |
 |
A057175 | 3, 59, 223, 547, 773, 1009, 1823, ... |
 |
A057173 | 3, 17, 23, 29, 47, 61, 1619, ... |
 |
A057172 | 3, 11, 31, 43, 47, 59, 107, ... |
 |
A057171 | 5, 67, 101, 103, 229, 347, 4013, ... |
 |
A007658 | 3, 5, 7, 13, 23, 43, 281, ... |
 |
A000978 | 3, 5, 7, 11, 13, 17, 19, ... |
| 2 | A000043 | 2, 3, 5, 7, 13, 17, 19, 31, 61, 89, 107, ... |
| 3 | A028491 | 3, 7, 13, 71, 103, 541, 1091, 1367, ... |
| 5 | A004061 | 3, 7, 11, 13, 47, 127, 149, 181, 619, ... |
| 6 | A004062 | 2, 3, 7, 29, 71, 127, 271, 509, 1049, ... |
| 7 | A004063 | 5, 13, 131, 149, 1699, ... |
| 10 | A004023 | 2, 19, 23, 317, 1031, ... |
| 11 | A005808 | 17, 19, 73, 139, 907, 1907, 2029, 4801, ... |
| 12 | A004064 | 2, 3, 5, 19, 97, 109, 317, 353, 701, ... |
Yates (1982) published all the repunit factors for 
, a portion of which are reproduced in the Wolfram Language notebook by Weisstein. Brillhart et al. (1988) gave a table of repunit factors which cannot be obtained algebraically, and a continuously updated version of this table is now maintained online. These tables include factors for 
(with 
odd) and 
(with 
even and odd). After algebraically factoring 
, these types of factors are sufficient for complete factorizations.
The sequence of least 
such that 
is prime for 
, 2, ... are 2, 3, 2, 3, 2, 5, 3, 0, 2, 17, 2, 5, ... (OEIS A084740), and the sequence of least 
such that 
is prime for 
, 2, ... are 3, 3, 3, 5, 3, 3, 0, 3, 5, 5, 5, 3, ... (OEIS A084742).
A Smith number can be constructed from every factored repunit.
REFERENCES:
Ball, W. W. R. and Coxeter, H. S. M. Mathematical Recreations and Essays, 13th ed. New York: Dover, p. 66, 1987.
Baxter, L. "R86453 Is a New Probable Prime Repunit." 26 Oct 2000. https://listserv.nodak.edu/scripts/wa.exe?A2=ind0010&L=nmbrthry&P=2557.
Beiler, A. H. "11111...111." Ch. 11 in Recreations in the Theory of Numbers: The Queen of Mathematics Entertains. New York: Dover, 1966.
Brillhart, J.; Lehmer, D. H.; Selfridge, J.; Wagstaff, S. S. Jr.; and Tuckerman, B. Factorizations of b-n+/-1, b=2, 3, 5, 6, 7, 10, 11, 12 Up to High Powers, rev. ed. Providence, RI: Amer. Math. Soc., 1988.
Di Maria, G. "The Repunit Primes Project." https://www.repunit.org/.
Dubner, H. "Generalized Repunit Primes." Math. Comput. 61, 927-930, 1993.
Dubner, H. "New prp Repunit R(49081)." 9 Sep 1999. https://listserv.nodak.edu/scripts/wa.exe?A2=ind9909&L=nmbrthry&P=740.
Dubner, H. "Repunit 
is a Probable Prime." Math. Comput. 71, 833-835, 2002. https://www.ams.org/mcom/2002-71-238/.
Dubner, H. "New Repunit R(109297)." 3 Apr 2007. https://listserv.nodak.edu/cgi-bin/wa.exe?A2=ind0704&L=nmbrthry&T=0&P=178.
Dubner, H. and Granlund, T. "Primes of the Form 
." J. Int. Sequences 3, No. 00.2.7, 2000. https://www.cs.uwaterloo.ca/journals/JIS/VOL3/DUBNER/dubner.html.
Dudeney, H. E. The Canterbury Puzzles and Other Curious Problems, 7th ed. London: Thomas Nelson and Sons, 1949.
Gardner, M. The Sixth Book of Mathematical Games from Scientific American. Chicago, IL: University of Chicago Press, pp. 85-86, 1984.
Granlund, T. "Repunits." https://www.swox.com/gmp/repunit.html.
Guy, R. K. "Mersenne Primes. Repunits. Fermat Numbers. Primes of Shape 
." §A3 in Unsolved Problems in Number Theory, 2nd ed. New York: Springer-Verlag, pp. 8-13, 1994.
Madachy, J. S. Madachy's Mathematical Recreations. New York: Dover, pp. 152-153, 1979.
Ribenboim, P. "Repunits and Similar Numbers." §5.5 in The New Book of Prime Number Records. New York: Springer-Verlag, pp. 350-354, 1996.
Sloane, N. J. A. Sequences A000043/M0672, A000225/M2655, A000978, A001562, A002275, A002477/M5386, A002450/M3914, A002452/M4733, A003462/M3463, A007583, A007658, A003463/M4209, A003464/M4425, A004023/M2114, A004023/M2114, A004061/M2620, A004062/M0861, A004063/M3836, A004064/M0744, A005808/M5032, A016123, A016125, A023000, A023001, A028491/M2643, A046053, A057171, A057172, A057173, A057175, A057177, A057178, A066443, A084740, and A084742 in "The On-Line Encyclopedia of Integer Sequences."
Snyder, W. M. "Factoring Repunits." Am. Math. Monthly 89, 462-466, 1982.
Sorli, R. "Factorization Tables." https://www-staff.maths.uts.edu.au/~rons/fact/fact.htm.
Voznyy, M. and Budnyy, A. "New PRP Repunit R(270343)." 15 Jul 2007. https://listserv.nodak.edu/cgi-bin/wa.exe?A2=ind0707&L=nmbrthry&T=0&P=1086.
Williams, H. C. and Dubner, H. "The Primality of 
." Math. Comput. 47, 703-711, 1986.
Williams, H. C. and Seah, E. "Some Primes of the Form 
. Math. Comput. 33, 1337-1342, 1979.
Yates, S. "Peculiar Properties of Repunits." J. Recr. Math. 2, 139-146, 1969.
Yates, S. "Prime Divisors of Repunits." J. Recr. Math. 8, 33-38, 1975.
Yates, S. "The Mystique of Repunits." Math. Mag. 51, 22-28, 1978.
Yates, S. Repunits and Reptends. Delray Beach, FL: S. Yates, 1982.
الاكثر قراءة في نظرية الاعداد
اخر الاخبار
اخبار العتبة العباسية المقدسة
الآخبار الصحية
قسم الشؤون الفكرية يصدر كتاباً يوثق تاريخ السدانة في العتبة العباسية المقدسة
"المهمة".. إصدار قصصي يوثّق القصص الفائزة في مسابقة فتوى الدفاع المقدسة للقصة القصيرة
(نوافذ).. إصدار أدبي يوثق القصص الفائزة في مسابقة الإمام العسكري (عليه السلام)
