Necessary and Sufficient Conditions for Palindromic Numbers to Be Expressible as Sums of a Number and Its Reverse: The Divisibility Theorem by 11 and Its Application in Structural Analysis of Lychrel Numbers
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2026
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| _version_ | 1866901351580041216 |
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| author | li, hong |
| author_facet | li, hong |
| contents | <p>This paper investigates the divisibility properties and representability of palindromic numbers with arbitrary digit lengths. It proves that even-digit palindromic numbers are always divisible by 11, and establishes the necessary and sufficient condition for odd-digit palindromic numbers to be divisible by 11. A core theorem is constructed: a palindromic number can be expressed as the sum of a natural number and its reverse (with leading zeros allowed for digit alignment) if and only if it is divisible by 11. The conclusion is generalized to numbers of any digit length, and the inherent contradiction that renders non-multiples of 11 unrepresentable is explained through carryover structure. Iterative verification is performed on the 196 family of Lychrel candidates, revealing the structural root of their failure to produce palindromic numbers. The results of this paper provide a rigorous and unified criterion for the construction and determination of palindromic numbers in elementary number theory.</p> <p>CORRECTION NOTICE: This manuscript contains critical errors and is hereby retracted in its current form. The authors apologize for any inconvenience caused</p> <p> </p> |
| format | Recurso digital |
| id | zenodo_https___doi_org_10_5281_zenodo_19395252 |
| institution | Zenodo |
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| publishDate | 2026 |
| publisher | Zenodo |
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| spellingShingle | Necessary and Sufficient Conditions for Palindromic Numbers to Be Expressible as Sums of a Number and Its Reverse: The Divisibility Theorem by 11 and Its Application in Structural Analysis of Lychrel Numbers li, hong <p>This paper investigates the divisibility properties and representability of palindromic numbers with arbitrary digit lengths. It proves that even-digit palindromic numbers are always divisible by 11, and establishes the necessary and sufficient condition for odd-digit palindromic numbers to be divisible by 11. A core theorem is constructed: a palindromic number can be expressed as the sum of a natural number and its reverse (with leading zeros allowed for digit alignment) if and only if it is divisible by 11. The conclusion is generalized to numbers of any digit length, and the inherent contradiction that renders non-multiples of 11 unrepresentable is explained through carryover structure. Iterative verification is performed on the 196 family of Lychrel candidates, revealing the structural root of their failure to produce palindromic numbers. The results of this paper provide a rigorous and unified criterion for the construction and determination of palindromic numbers in elementary number theory.</p> <p>CORRECTION NOTICE: This manuscript contains critical errors and is hereby retracted in its current form. The authors apologize for any inconvenience caused</p> <p> </p> |
| title | Necessary and Sufficient Conditions for Palindromic Numbers to Be Expressible as Sums of a Number and Its Reverse: The Divisibility Theorem by 11 and Its Application in Structural Analysis of Lychrel Numbers |
| url | https://doi.org/10.5281/zenodo.19395252 |