61

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Summary

Factorization

The number 61 is a prime number.

Properties and families

Property or family	Parameter values	First few numbers	Proof of satisfaction/membership/containment
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		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

Waring representations

Sums of squares

Item	Value
unique (up to plus/minus and ordering) representation as sum of two squares	$6^{2}+5^{2}$. Note that existence and uniqueness both follow from it being a prime that is 1 mod 4. This also corresponds to the factorization $61=(6+5i)(6-5i)$ in the ring of Gaussian integers $\mathbb {Z} [i]$.
representations as sum of three squares (up to ordering and plus/minus equivalence)	$6^{2}+5^{2}+0^{2}$ $6^{2}+4^{2}+3^{2}$

Prime-generating polynomials

Below are some polynomials that give prime numbers for small input values, which give the value 61 for suitable input choice.

Polynomial	Degree	Some values for which it generates primes	Input value $n$ at which it generates 61
$n^{2}-n+41$	2	all numbers 1-40, because 41 is one of the lucky numbers of Euler.	5
$2n^{2}+29$	2	all numbers 0-28	4