| _version_ | 1866901181545054208 |
|---|---|
| author | Coronato, Antonio |
| author_facet | Coronato, Antonio |
| contents | <p>Every even–digit base–10 palindrome is divisible by 11, but the structure of its</p> <p>quotient is far from trivial. Let. P= d1d2 . . . dndn . . . d2d1</p> <p>be such a palindrome and Q= P/11. Empirically, Q is often a palindrome; when it is</p> <p>not, it differs from a palindrome by a remarkably small correction involving powers of</p> <p>10. We give a structural explanation of this phenomenon.</p> <p>We show that every such quotient admits a decomposition</p> <p>Q= Qideal + ∆</p> <p>where Qideal is a (2n−1)-digit palindrome determined by a unique symmetric solution</p> <p>of the carry equations in</p> <p>P= Q + 10Q,</p> <p>and ∆ is a finite sum of elementary corrections induced by borrow chains: maximal</p> <p>intervals where the actual carry pattern deviates from its symmetric counterpart. We</p> <p>prove that the number of borrow chains equals the number of monotonicity reversals</p> <p>in (d1, . . . , dn), and that each chain contributes either a term ±10k or ±(10k</p> <p>−10k−m).</p> <p>A computational test over 100,000 random palindromes confirms the theory without</p> <p>exceptions.</p> |
| format | Recurso digital |
| id | zenodo_https___doi_org_10_5281_zenodo_17622188 |
| institution | Zenodo |
| language | |
| publishDate | 2025 |
| publisher | Zenodo |
| record_format | zenodo |
| spellingShingle | On the structure of the quotient of even-digital base-10 palindrome divisible by 11 Coronato, Antonio <p>Every even–digit base–10 palindrome is divisible by 11, but the structure of its</p> <p>quotient is far from trivial. Let. P= d1d2 . . . dndn . . . d2d1</p> <p>be such a palindrome and Q= P/11. Empirically, Q is often a palindrome; when it is</p> <p>not, it differs from a palindrome by a remarkably small correction involving powers of</p> <p>10. We give a structural explanation of this phenomenon.</p> <p>We show that every such quotient admits a decomposition</p> <p>Q= Qideal + ∆</p> <p>where Qideal is a (2n−1)-digit palindrome determined by a unique symmetric solution</p> <p>of the carry equations in</p> <p>P= Q + 10Q,</p> <p>and ∆ is a finite sum of elementary corrections induced by borrow chains: maximal</p> <p>intervals where the actual carry pattern deviates from its symmetric counterpart. We</p> <p>prove that the number of borrow chains equals the number of monotonicity reversals</p> <p>in (d1, . . . , dn), and that each chain contributes either a term ±10k or ±(10k</p> <p>−10k−m).</p> <p>A computational test over 100,000 random palindromes confirms the theory without</p> <p>exceptions.</p> |
| title | On the structure of the quotient of even-digital base-10 palindrome divisible by 11 |
| url | https://doi.org/10.5281/zenodo.17622188 |