1105 (number)

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Cardinal one thousand one hundred five
Ordinal 1105th
(one thousand one hundred fifth)
Factorization 5 × 13 × 17
Greek numeral ,ΑΡΕ´
Roman numeral MCV
Binary 100010100012
Ternary 11112213
Senary 50416
Octal 21218
Duodecimal 78112
Hexadecimal 45116

1105 (eleven hundred [and] five, or one thousand one hundred [and] five) is the natural number following 1104 and preceding 1106.

1105 is the smallest positive integer that is a sum of two positive squares in exactly four different ways, [1] [2] a property that can be connected (via the sum of two squares theorem) to its factorization 5 × 13 × 17 as the product of the three smallest prime numbers that are congruent to 1 modulo 4. [2] [3] It is also the smallest member of a cluster of three semiprimes (1105, 1106, 1107) with eight divisors, [4] and the second-smallest Carmichael number, after 561, [5] [6] one of the first four Carmichael numbers identified by R. D. Carmichael in his 1910 paper introducing this concept. [6] [7]

Its binary representation 10001010001 and its base-4 representation 101101 are both palindromes, [8] and (because the binary representation has nonzeros only in even positions and its base-4 representation uses only the digits 0 and 1) it is a member of the Moser–de Bruijn sequence of sums of distinct powers of four. [9]

As a number of the form for 13, 1105 is the magic constant for 13 × 13 magic squares, [10] and as a difference of two consecutive fourth powers (1105 = 74− 64) [11] [12] it is a rhombic dodecahedral number (a type of figurate number), and a magic number for body-centered cubic crystals. [11] [13] These properties are closely related: the difference of two consecutive fourth powers is always a magic constant for an odd magic square whose size is the sum of the two consecutive numbers (here 7 + 6 = 13). [11]

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References

  1. ↑ Sloane, N. J. A. (ed.). "SequenceA016032(Least positive integer that is the sum of two squares of positive integers in exactly n ways)". The On-Line Encyclopedia of Integer Sequences . OEIS Foundation.
  2. 1 2 Tenenbaum, Gérald (1997). "1105: first steps in a mysterious quest". In Graham, Ronald L.; Nešetřil, Jaroslav (eds.). The mathematics of Paul Erdős, I. Algorithms and Combinatorics. Vol. 13. Berlin: Springer. pp. 268–275. doi:10.1007/978-3-642-60408-9_21. ISBN   978-3-642-64394-1. MR   1425191.
  3. ↑ Sloane, N. J. A. (ed.). "SequenceA006278(product of the first n primes congruent to 1 (mod 4))". The On-Line Encyclopedia of Integer Sequences . OEIS Foundation.
  4. ↑ Sloane, N. J. A. (ed.). "SequenceA005238(Numbers k such that k, k+1 and k+2 have the same number of divisors)". The On-Line Encyclopedia of Integer Sequences . OEIS Foundation.
  5. ↑ Sloane, N. J. A. (ed.). "SequenceA002997(Carmichael numbers)". The On-Line Encyclopedia of Integer Sequences . OEIS Foundation.
  6. 1 2 Křížek, Michal; Luca, Florian; Somer, Lawrence (2001). 17 Lectures on Fermat Numbers: From Number Theory to Geometry. CMS Books in Mathematics/Ouvrages de Mathématiques de la SMC. Vol. 9. Springer-Verlag, New York. p. 136. doi:10.1007/978-0-387-21850-2. ISBN   0-387-95332-9. MR   1866957.
  7. ↑ Carmichael, R. D. (1910). "Note on a new number theory function". Bulletin of the American Mathematical Society. 16 (5): 232–238. doi: 10.1090/S0002-9904-1910-01892-9 . JFM   41.0226.04.
  8. ↑ Sloane, N. J. A. (ed.). "SequenceA097856(Numbers that are palindromic in bases 2 and 4)". The On-Line Encyclopedia of Integer Sequences . OEIS Foundation.
  9. ↑ Sloane, N. J. A. (ed.). "SequenceA000695(Moser-de Bruijn sequence)". The On-Line Encyclopedia of Integer Sequences . OEIS Foundation.
  10. ↑ Sloane, N. J. A. (ed.). "SequenceA006003". The On-Line Encyclopedia of Integer Sequences . OEIS Foundation.
  11. 1 2 3 Sloane, N. J. A. (ed.). "SequenceA005917(Rhombic dodecahedral numbers)". The On-Line Encyclopedia of Integer Sequences . OEIS Foundation.
  12. ↑ Gould, H. W. (1978). "Euler's formula for th differences of powers". The American Mathematical Monthly . 85 (6): 450–467. doi:10.1080/00029890.1978.11994613. JSTOR   2320064. MR   0480057.
  13. ↑ Jiang, Aiqin; Tyson, Trevor A.; Axe, Lisa (September 2005). "The structure of small Ta clusters". Journal of Physics: Condensed Matter. 17 (39): 6111–6121. Bibcode:2005JPCM...17.6111J. doi:10.1088/0953-8984/17/39/001. S2CID   41954369.
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