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Zenodo
2026
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| Online-Zugang: | https://doi.org/10.5281/zenodo.18807582 |
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| _version_ | 1866901243392163840 |
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| author | Brady, Kyle |
| author_facet | Brady, Kyle |
| contents | <p>Building on Brady’s Theorem, which established the two foundational zero-middle-digit families covering every base not divisible by three, five infinite families of three-digit narcissistic numbers are now known. Family 1 (b = 3n + 1) and Family 2 (b = 3n + 2) are the direct content of Brady’s Theorem, together guaranteeing a narcissistic number in every base b ≥ 4 with 3 ∤ b — producing, among others, 407 and 370 as their respective base-10 and base-11 representatives. Family 3 (b = 6n + 4) answers the open question Brady’s paper posed about nonzero middle digits: its formula (b − 1)(b² + 2) / 6 yields 153 at b = 10, the smallest three-digit narcissistic number in base 10, and extends to infinitely many bases beyond it. Family 4 (b = 15n + 1) operates on a finer mod-15 structure entirely independent of the mod-3 framework, and notably produces 371 at base 16 — the one base-10 narcissistic number that has no family membership in base 10 itself. Family 5 (b = 9n + 7) generates trailing-zero narcissistic numbers via the elegant formula b × (111_b / 3), and its formula evaluated at b = 10 produces 370, even though b = 10 lies outside the family’s proven base condition — a tantalising hint that the domain may admit further generalisation. Together, the five families account for three of the four base-10 narcissistic numbers structurally, leave 371 as the only base-10 sporadic case, and collectively raise the question of whether any infinite family remains undiscovered.</p> |
| format | Recurso digital |
| id | zenodo_https___doi_org_10_5281_zenodo_18807582 |
| institution | Zenodo |
| language | |
| publishDate | 2026 |
| publisher | Zenodo |
| record_format | zenodo |
| spellingShingle | Brady's Theorem Families Brady, Kyle <p>Building on Brady’s Theorem, which established the two foundational zero-middle-digit families covering every base not divisible by three, five infinite families of three-digit narcissistic numbers are now known. Family 1 (b = 3n + 1) and Family 2 (b = 3n + 2) are the direct content of Brady’s Theorem, together guaranteeing a narcissistic number in every base b ≥ 4 with 3 ∤ b — producing, among others, 407 and 370 as their respective base-10 and base-11 representatives. Family 3 (b = 6n + 4) answers the open question Brady’s paper posed about nonzero middle digits: its formula (b − 1)(b² + 2) / 6 yields 153 at b = 10, the smallest three-digit narcissistic number in base 10, and extends to infinitely many bases beyond it. Family 4 (b = 15n + 1) operates on a finer mod-15 structure entirely independent of the mod-3 framework, and notably produces 371 at base 16 — the one base-10 narcissistic number that has no family membership in base 10 itself. Family 5 (b = 9n + 7) generates trailing-zero narcissistic numbers via the elegant formula b × (111_b / 3), and its formula evaluated at b = 10 produces 370, even though b = 10 lies outside the family’s proven base condition — a tantalising hint that the domain may admit further generalisation. Together, the five families account for three of the four base-10 narcissistic numbers structurally, leave 371 as the only base-10 sporadic case, and collectively raise the question of whether any infinite family remains undiscovered.</p> |
| title | Brady's Theorem Families |
| url | https://doi.org/10.5281/zenodo.18807582 |