# Idempotent Factorizations of Square-Free Integers

## Abstract

**:**

## 1. Introduction

## 2. Idempotent Factorizations of a Carmichael Number

## 3. Maximally Idempotent Integers

**Theorem**

**1.**

**Theorem**

**2.**

## 4. Strong Impostors and Idempotent Factorizations

## 5. Examples

#### Cumulative Statistics for Idempotent Factorizations of the Carmichael Numbers

## 6. Constructing Maximally Idempotent Integers

## 7. Cumulative Statistics on Idempotent Factorizations

## 8. Idempotent Tuples and RSA

**Theorem**

**3.**

**Proof.**

## 9. Conclusions and Future Work

## Acknowledgments

## Conflicts of Interest

## References

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n | p or $\overline{\mathit{p}}$ | $\overline{\mathit{q}}$ |
---|---|---|

30 | 5 | 6 |

42 | 7 | 6 |

66 | 11 | 6 |

78 | 13 | 6 |

102 | 17 | 6 |

105 | 7 | 15 |

114 | 19 | 6 |

130 | 13 | 10 |

138 | 23 | 6 |

165 | 11 | 15 |

170 | 17 | 10 |

174 | 29 | 6 |

182 | 13 | 14 |

186 | 31 | 6 |

195 | 13 | 15 |

210 | 10 | 21 |

3 Factors | $\mathit{\lambda}$ | 4 Factors | $\mathit{\lambda}$ | 5 Factors | $\mathit{\lambda}$ |
---|---|---|---|---|---|

273 = $3\times 7\times 13$ | 12 | $\underline{\mathrm{63,973}}=7\times 13\times 19\times 37$ | 36 | $\mathrm{72,719,023}=13\times 19\times 37\times 73\times 109$ | 216 |

$455=5\times 7\times 13$ | 12 | $\mathrm{137,555}=5\times 11\times 41\times 61$ | 120 | $\mathrm{213,224,231}=11\times 31\times 41\times 101\times 151$ | 300 |

$\underline{1729}=7\times 13\times 19$ | 36 | $\mathrm{145,607}=7\times 11\times 31\times 61$ | 60 | ||

$2109=3\times 19\times 37$ | 36 | $\mathrm{245,791}=7\times 13\times 37\times 73$ | 72 | ||

$2255=5\times 11\times 41$ | 40 | $\mathrm{356,595}=5\times 19\times 37\times 73$ | 72 | ||

$2387=7\times 11\times 31$ | 30 | $\mathrm{270,413}=11\times 13\times 31\times 61$ | 60 | ||

$3367=7\times 13\times 37$ | 36 | $\mathrm{536,389}=7\times 19\times 37\times 109$ | 108 | ||

$3515=5\times 19\times 37$ | 72 | $\mathrm{667,147}=13\times 19\times 37\times 73$ | 72 | ||

$4433=11\times 13\times 31$ | 60 | $\mathrm{996,151}=13\times 19\times 37\times 109$ | 108 | ||

$4697=7\times 11\times 61$ | 60 | $\mathrm{1,007,903}=13\times 31\times 41\times 61$ | 120 | ||

$4921=7\times 19\times 37$ | 36 | $\mathrm{1,847,747}=11\times 17\times 41\times 241$ | 240 | ||

$5673=3\times 31\times 61$ | 60 | $\mathrm{1,965,379}=13\times 19\times 73\times 109$ | 216 | ||

$6643=7\times 13\times 73$ | 72 | $\mathrm{2,060,863}=7\times 37\times 73\times 109$ | 216 | ||

$6935=5\times 19\times 73$ | 72 | $\mathrm{2,395,897}=7\times 31\times 61\times 181$ | 180 | ||

$7667=11\times 17\times 41$ | 80 | $\mathrm{2,778,611}=11\times 41\times 61\times 101$ | 600 | ||

$8103=3\times 37\times 73$ | 72 | $\mathrm{3,140,951}=11\times 31\times 61\times 151$ | 300 |

Proportion of Maximally Idempotent Integers | |||
---|---|---|---|

# factors | Carmichael #’s $<{10}^{18}$ | integers $<{2}^{30}$ | ratio |

3 | $5.5862\times {10}^{-4}$ | $1.4145\times {10}^{-5}$ | 39.5 |

4 | $2.3543\times {10}^{-5}$ | $2.9336\times {10}^{-7}$ | 80.3 |

5 | $7.1344\times {10}^{-7}$ | $1.8626\times {10}^{-9}$ | 383.0 |

p | $\mathit{\mu}\left(\mathit{p}\right)$ |
---|---|

13 | 2 |

31 | 1 |

37 | 7 |

41 | 1 |

61 | 11 |

67 | 1 |

73 | 14 |

89 | 1 |

97 | 2 |

109 | 9 |

113 | 2 |

127 | 2 |

156 | 11 |

181 | 19 |

193 | 8 |

199 | 3 |

Max n | ${2}^{12}$ | ${2}^{15}$ | ${2}^{18}$ | ${2}^{21}$ | ${2}^{24}$ | ${2}^{27}$ | ${2}^{30}$ |
---|---|---|---|---|---|---|---|

${R}_{sf}$ | 0.61 | 0.37 | 0.28 | 0.21 | 0.17 | 0.13 | 0.11 |

${R}_{N}$ | 0.09 | 0.09 | 0.08 | 0.07 | 0.06 | 0.05 | 0.04 |

${R}_{cpu}$ | - | 2.7 | 11.3 | 10.6 | 13.3 | 9.8 | 10.4 |

# Factors | 0 | 1 | 2 | 3 | 4 | 5 | 6 | 7 |
---|---|---|---|---|---|---|---|---|

3 | 184,510,285 | 34,215,577 | 0 | 15,189 | 0 | 0 | 0 | 0 |

4 | 132,479,584 | 11,347,214 | 4448 | 15,678 | 28 | 235 | 0 | 315 |

5 | 50,515,758 | 1,733,232 | 6530 | 13,743 | 93 | 599 | 1 | 441 |

6 | 10,004,651 | 242,377 | 6143 | 6906 | 167 | 586 | 12 | 302 |

7 | 931,270 | 35,473 | 2994 | 1597 | 124 | 286 | 22 | 102 |

8 | 29,211 | 2956 | 477 | 158 | 39 | 43 | 5 | 6 |

9 | 99 | 28 | 7 | 2 | 1 | 0 | 1 | 1 |

# Factors | ||||||
---|---|---|---|---|---|---|

5 | 8:2 | 9:6 | 11:18 | 15:2 | ||

6 | 8:3 | 9:10 | 11:31 | 15:20 | ||

7 | 8:3 | 9:5 | 10:1 | 11:24 | 15:3 | 31:1 |

8 | 8:1 | 9:2 | 11:4 |

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**MDPI and ACS Style**

Fagin, B.
Idempotent Factorizations of Square-Free Integers. *Information* **2019**, *10*, 232.
https://doi.org/10.3390/info10070232

**AMA Style**

Fagin B.
Idempotent Factorizations of Square-Free Integers. *Information*. 2019; 10(7):232.
https://doi.org/10.3390/info10070232

**Chicago/Turabian Style**

Fagin, Barry.
2019. "Idempotent Factorizations of Square-Free Integers" *Information* 10, no. 7: 232.
https://doi.org/10.3390/info10070232