41: Difference between revisions

Revision as of 04:15, 16 January 2012

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The number 41 is a prime number.

Property or family	Parameter values	First few numbers	Proof of satisfaction/membership/containment
prime number	it is the 12th prime number	2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97, [SHOW MORE] 101, 103, 107, 109, 113, 127, 131, 137, 139, 149, 151, 157, 163, 167, 173, 179, 181, 191, 193, 197, 199, 211, 223, 227, 229, 233, 239, 241, 251, 257, 263, 269, 271 View list on OEIS	divide and check
regular prime	first regular prime occurring after an irregular prime	3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 41, 43, 47, 53, 61, [SHOW MORE] 71, 73, 79, 83, 89, 97, 107, 109, 113, 127, 137, 139, 151, 163, 167, 173, 179, 181, 191, 193, 197, 199, 211, 223, 227, 229, 239, 241, 251, 269, 277, 281, 313, 317, 331, 337, 349, 359, 367, 373, 383, 397, 419, 431 View list on OEIS
lucky number of Euler	biggest of six such numbers	2, 3, 5, 11, 17, 41 View list on OEIS	A prime $p$ is a lucky number of Euler iff the ring of integers in $\mathbb {Q} ({\sqrt {1-4p}})$ is a unique factorization domain.

Item	Value
unique (up to plus/minus and ordering) representation as sum of two squares	$5^{2}+4^{2}$ . Note that existence and uniqueness both follow from it being a prime that is 1 mod 4. This also corresponds to the factorization $17=(5+4i)(5-4i)$ in the ring of Gaussian integers $\mathbb {Z} [i]$ .
representations as sum of three squares (up to ordering and plus/minus equivalence)	$5^{2}+4^{2}+0^{2}$ $4^{2}+4^{2}+3^{2}$ $6^{2}+2^{2}+1^{2}$

Below are some polynomials that give prime numbers for small input values, which give the value 41 for suitable input choice.

Polynomial	Degree	Some values for which it generates primes	Input value $n$ at which it generates 41
$n^{2}-n+11$	2	all numbers 1-10, because 11 is one of the lucky numbers of Euler.	6
$n^{2}-n+41$	2	all numbers 1-40, because 41 is one of the lucky numbers of Euler.	1

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 | [[satisfies property::lucky number of Euler]] || biggest of six such numbers || {{#lst:lucky number of Euler|list}} || A prime <math>p</math> is a lucky number of Euler iff the ring of integers in <math>\mathbb{Q}(\sqrt{1 - 4p})</math> is a unique factorization domain.
+|}
+==Waring representations==
+===Sums of squares===
+{{square sums facts to check against}}
+{| class="sortable" border="1"
+! Item !! Value
+|-
+| unique (up to plus/minus and ordering) representation as sum of two squares || <math>5^2 + 4^2</math>. Note that existence and uniqueness both follow from it being a prime that is 1 mod 4.<br>This also corresponds to the factorization <math>17 = (5+ 4i)(5 - 4i)</math> in the [[ring of Gaussian integers]] <math>\mathbb{Z}[i]</math>.
+|-
+| representations as sum of three squares (up to ordering and plus/minus equivalence) || <math>5^2 + 4^2 + 0^2</math><br><math>4^2 + 4^2 + 3^2</math><br><math>6^2 + 2^2 + 1^2</math>
 |}