1105 (eleven hundred
ndfive, or one thousand one hundred
ndfive) is the
natural number
In mathematics, the natural numbers are the numbers 0, 1, 2, 3, and so on, possibly excluding 0. Some start counting with 0, defining the natural numbers as the non-negative integers , while others start with 1, defining them as the positive in ...
following 1104 and preceding 1106.
Mathematical properties
1105 is the smallest positive integer that is a sum of two positive squares in exactly four different ways,
[ a property that can be connected (via the ]sum of two squares theorem
In number theory, the sum of two squares theorem relates the prime decomposition of any integer to whether it can be written as a sum of two Square number, squares, such that for some integers , .
An integer greater than one can be written as a ...
) to its factorization as the product of the three smallest prime number
A prime number (or a prime) is a natural number greater than 1 that is not a Product (mathematics), product of two smaller natural numbers. A natural number greater than 1 that is not prime is called a composite number. For example, 5 is prime ...
s that are congruent to 1 modulo 4.semiprime
In mathematics, a semiprime is a natural number that is the product of exactly two prime numbers. The two primes in the product may equal each other, so the semiprimes include the squares of prime numbers.
Because there are infinitely many prime n ...
s (1105, 1106, 1107) with eight divisor
In mathematics, a divisor of an integer n, also called a factor of n, is an integer m that may be multiplied by some integer to produce n. In this case, one also says that n is a '' multiple'' of m. An integer n is divisible or evenly divisibl ...
s, and the second-smallest Carmichael number
In number theory, a Carmichael number is a composite number which in modular arithmetic satisfies the congruence relation:
: b^n\equiv b\pmod
for all integers . The relation may also be expressed in the form:
: b^\equiv 1\pmod
for all integers b ...
, after 561
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Year 561 ( DLXI) was a common year starting on Saturday of the Julian calendar. The denomination 561 for this year has been used since the early medieval period, when the Anno Domini calendar era became the prevalent method in Europe ...
,[
Its ]binary representation
A binary number is a number expressed in the base-2 numeral system or binary numeral system, a method for representing numbers that uses only two symbols for the natural numbers: typically "0" (zero) and "1" (one). A ''binary number'' may also ...
10001010001 and its base-4 representation 101101 are both palindromes
A palindrome ( /ˈpæl.ɪn.droʊm/) is a word, number, phrase, or other sequence of symbols that reads the same backwards as forwards, such as ''madam'' or '' racecar'', the date " 02/02/2020" and the sentence: "A man, a plan, a canal – Pana ...
, 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
In number theory, the Moser–de Bruijn sequence is an integer sequence named after Leo Moser and Nicolaas Govert de Bruijn, consisting of the sums of distinct powers of 4. Equivalently, they are the numbers whose binary representations are no ...
of sums of distinct powers of four.
As a number of the form for 1105 is the magic constant
The magic constant or magic sum of a magic square is the sum of numbers in any row, column, or diagonal of the magic square. For example, the magic square shown below has a magic constant of 15. For a normal magic square of order ''n'' – that is ...
for magic square
In mathematics, especially History of mathematics, historical and recreational mathematics, a square array of numbers, usually positive integers, is called a magic square if the sums of the numbers in each row, each column, and both main diago ...
s, and as a difference of two consecutive fourth powers [ it is a rhombic dodecahedral number (a type of ]figurate number
The term figurate number is used by different writers for members of different sets of numbers, generalizing from triangular numbers to different shapes (polygonal numbers) and different dimensions (polyhedral numbers). The ancient Greek mathemat ...
), and a magic number for body-centered cubic crystals.[
]
References
{{reflist
Integers