# 61

From Number

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## Contents

## 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]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]View list on OEIS |

## Waring representations

### Sums of squares

Template:Square sums facts to check against

Item | Value |
---|---|

unique (up to plus/minus and ordering) representation as sum of two squares | . Note that existence and uniqueness both follow from it being a prime that is 1 mod 4. This also corresponds to the factorization in the ring of Gaussian integers . |

representations as sum of three squares (up to ordering and plus/minus equivalence) | |

## 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 at which it generates 61 |
---|---|---|---|

2 | all numbers 1-40, because 41 is one of the lucky numbers of Euler. | 5 | |

2 | all numbers 0-28 | 4 |