NONE 20260610053845096 wseas/wseas content-registration-form+ja@crossref.org content-registration-form EQUATIONS 2944-9146 2732-9976 Potůček R. Department of Mathematics and Physics, University of Defence, Kounicova 65, 662 10 Brno, CZECH REPUBLIC https://orcid.org/0000-0003-4385-691X We study positive integers whose binary representation is a palindrome of the form (1 . . . 1)(0 . . . 0)(1 . . . 1), where the number of ones is twice the integer a ≥ 1, and the number of zeros is b ≥ 0. We give an elementary algebraic description of all such integers, derive a factorization formula, and deduce consequences for primality. In particular we show that except for the case a = 1 (which yields the numbers 2 m + 1 with m = b + 1), every such integer is composite by a simple factorization; consequently the only possible primes in this family are Fermat-type numbers 2 m + 1, and thus (by a classical necessary condition) their exponents m must be powers of two. Several illustrative examples are given and implications for search strategies are discussed. 12 31 2025 12 31 2025 100 10 https://creativecommons.org/licenses/by/4.0/deed.en_US 10.37394/232021.2025.5.10 https://wseas.com/journals/equations/2025/a20equations-009(2025).pdf Křížek, M, Luca, F., Somer, L. 17 Lectures on Fermat Numbers: From Number Theory to Geometry. Springer, New York, 2013. ISBN 978-0-387-95332-9. https://welib.org/md5/ 852618ee9d9a97ba9e71c94d4da1aa04 Brent, R. Three new factors of Fermat numbers. Mathematics of Computation, American Mathematical Society, Volume 69, Number 231, 2000, pp. 1297–1304. ISSN 1088-6842. https://doi.org/10.1090/S0025-5718-00-012072 Křížek, M, Somer, L. 17 necessary and sufficient conditions for the primality of Fermat numbers. Acta Mathematica et Informatica Universitatis Ostraviensis, Volume 11, Issue 1, 2003, pp. 73-79. ISSN 1211-4774. https://dml.cz/handle/10338.dmlcz/120594 Stein, W. Fermat Numbers. University of Washington, Math 414 Number Theory, study text, 2010, 24 pp. https://www.academia.edu/ 24443661/Fermat-Numbers Shevelev V., Garcia-Pulgarin G., Velásquez-Soto, J. M., Castillo, J. H. Overpseudoprimes, and Mersenne and Fermat numbers as primover numbers. Journal of Integer Sequences, Volume 15, 2012, Article 12.7.7, 10 pp. ISSN 1530-7638. https://cs.uwaterloo.ca/journals/JIS/VOL15/ Castillo/castillo2.html https://www.arxiv.org/abs/1206.0606 Cosgrave, J. A role for generalized Fermat numbers. Mathematics of Computation, American Mathematical Society, Volume 86, Number 304, 2016, pp. 899–933. ISSN 1088- 6842. https://doi.org/10.1090/MCOM/3111 Wang, X. Algorithm Available for Factoring Big Fermat Numbers. Journal of Software, Volume 15, Number 3, 2020, pp. 86-97. ISSN 1796- 217X. https://doi.org/10.17706/jsw.15.3.86-97 Křížek, M, Somer, L., Šolcová, A. From Great Discoveries in Number Theory to Applications. Springer Nature Switzerland AG, 2021. ISBN 978-3-030-83898-0. https://welib.org/md5/ 645ca2b72003c0a51d6727613af45b05 https://doi.org/10.1007/978-3-030-83899-7 Brändli G., Waldvogel, J. Six primes in generalized Fermat numbers. Elemente der Mathematik, Volume 80, 2025, pp. 59–65. ISSN 1420-8962. https://doi.org/10.4171/EM/522 Allouche, J.-P., Shallit, J. Sums of digits, overlaps, and palindromes. Discrete Mathematics and Theoretical Computer Science, Volume 4, 2000, pp. 1–10. ISSN 1365- 8050. https://www.academia.edu/1012617/ Sums-of-digits-overlaps-and-palindromes Anisiu, M.-C., Anisiu, V., Kása, z. Properties of palindromes in finite words. Pure Mathematics and Applications, Volume 17, Issue 3-4, 2006, pp. 183-195. ISSN 1218-4586. https://arxiv.org/abs/1002.2723 https://doi.org/10.48550/arXiv.1002.2723 Kása Z. Properties of palindromes. 2006, 13 pp. https://www.academia.edu/81346753/Propertiesof-palindromes Massé, A. B., Brlek, S., Frosini, A., Labbé, S., Rinaldi, S. Reconstructing words from a fixed palindromic length sequence. In: Ausiello, G., Karhumäki, J., Mauri, G., Ong, L. (eds) Fifth IFIP (International Federation for Information Processing) International Conference On Theoretical Computer Science, 2008, Volume 273. Springer, Boston, MA, pp. 101-114. https://link.springer.com/chapter/10.1007/978- 0-387-09680-3-7 Massé, A. B., Brlek, S., Garon, A., Labbé, S. Equations on palindromes and circular words. Theoretical Computer Science, Volume 412, Issue 27, 2011, pp. 2922-2930. ISSN 0304-3975. https://www.sciencedirect.com/ science/article/pii/S0304397510003816 https://doi.org/10.1016/j.tcs.2010.07.005 Antalan, J. R. M. Palindromes in Pythagorean Triple. 2015, 5 pp. https://www.academia.edu/ 11033751/Palindromes-in-Pythagorean-Triple Antalan, J. R. M. Numeric Palindromes in Primitive and Non-primitive Pythagorean Triples. JP Journal of Algebra, Number Theory and Applications, Volume 37, Issue 1, 2015, pp. 21-30. ISSN 0972-5555. https://doi.org/10.17654//JPANTAAug2015- 021-030 Dumitran, M., Gawrychowski, P., Manea, F. Longest Gapped Repeats and Palindromes. Discrete Mathematics and Theoretical Computer Science, Volume 19, Issue 4, 2017, 32 pp. ISSN 1365–8050. https://doi.org/10.1007/978-3-662-48057-1-16 https://arxiv.org/abs/1511.07180 Mahalingam, K., Sivasankar, M., Krithivasan, K. Palindromic Properties of Two-Dimensional Fibonacci Words. Romanian Journal of Information Science and Technology, Volume 21, Number 3, 2018, pp. 267–277. ISSN 1453- 8245. https://romjist.ro/full-texts/paper599.pdf Mahalingam, K. Counting scattered palindromes in a finite word. arXiv, Cornell University, 2021, 19 pp. https://www.academia.edu/143654198/Counting -scattered-palindromes-in-a-finite-word Gunarto, H., Islam, S. M. S., Majundar, A. A. K. Palindromes in Some SmarandacheType Functions. Jurnal Matematika MANTIK, Volume 8, Number 1, 2022, pp. 1-9. ISSN 2527- 3159. https://www.academia.edu/125621654/ Palindromes-in-Some-Smarandache-TypeFunctions Funakoshi, M., Mieno, T., Nakashima, Y., Inenaga, S., Bannai, H., Takeda, M. Computing palindromes and suffix trees of dynamic trees. Research Square, preprint, 2024, 22 pp. https://www.researchgate.net/publication/ 379427164-Computing-palindromesand-suffix-trees-of-dynamic-trees https://doi.org/10.21203/rs.3.rs-4167645/v1 Itzhaki, M. Palindromes Compression and Retrieval. Preprint, 2024, 12 pp. https:// www.researchgate.net/publication/384930427- Palindromes-Compression-and-Retrieval https://doi.org/10.48550/arXiv.2410.09984 Zverkov, O., Seliverstov, A., Shilovsky, G. Alignment of a Hidden Palindrome (in Russian). Mathematical Biology and Bioinformatics, Volume 19. Issue 2, 2024, pp. 427-438. https:// www.researchgate.net/publication/386750693- Alignment-of-a-Hidden-Palindrome https://doi.org/10.17537/2024.19.427 Mahalingam, K. Palindromes in adjacent factor swap of a word. Natural Computing, Volume 24, 2025, pp. 763-780. https://www.researchgate. net/publication/393900009-Palindromesin-adjacent-factor-swap-of-a-word https://doi.org/10.1007/s11047-025-10038-5 Ddamulira, M. Tribonacci-Lucas Numbers that are Palindromic Concatenations of Two Distinct Repdigits. Mathematica Pannonica, Volume 31, Issue 1, 2025, pp. 1-13. ISSN 0865-2090. https://doi.org/10.1556/314.2025.00020 Hardy, G. H., Wright, E. M. An Introduction to the Theory of Numbers. 6th revised Edition, Oxford University Press, London, 2008. ISBN 978-0-19-921986-5. https://www.scribd.com/document/419450907/ An-Introduction-to-the-Theory-of-Numbers https://books.google.cz/books/about/AnIntroduction-to-the-Theory-of-Numbers.html Ribenboim, P. The New Book of Prime Number Records. 3rd Edition, Springer, 1996. https://welib.org/md5/88f481bf436401c803092 2bedd3da0a2 Koblitz, N. A Course in Number Theory and Cryptography. 2nd Edition, Springer, 1994. ISBN 978-0-387-94293-9. https://welib.org/md5/86769a305f35c5a42d3f6 de186f57e55 Guy, R. K. Unsolved Problems in Number Theory. 4th Edition, Springer, 2004. ISBN 978-1-4419-1928-1. https://welib.org/md5/ ae46b2269bdb07ce0872a58b6db6da67 Wells, D. Prime Numbers: The Most Mysterious Figures in Math. John Wiley & Sons, 2005. ISBN 978-0-471-46234-7. https://welib.org/ md5/b80a20e95963876c33000d0f2d240e4e Sloane, N. J. A. The On-Line Encyclopedia of Integer Sequences, published electronically at https://oeis.org, 2025. Sequence A000215 (Fermat numbers: a(n) = 22 n + 1). https://oeis.org/A000215 Sloane, N. J. A. The On-Line Encyclopedia of Integer Sequences, published electronically at https://oeis.org, 2025. Sequence A006995 (Binary palindromes: numbers whose binary expansion is palindromic). https://oeis.org/A006995 Sloane, N. J. A. The On-Line Encyclopedia of Integer Sequences, published electronically at https://oeis.org, 2025. Sequence A002113 (Palindromes in base 10). https://oeis.org/A002113 Sloane, N. J. A. The On-Line Encyclopedia of Integer Sequences, published electronically at https://oeis.org, 2025. Sequence A002385 (Palindromic primes: prime numbers whose decimal expansion is a palindrome). https://oeis.org/A002385 Sloane, N. J. A. The On-Line Encyclopedia of Integer Sequences, published electronically at https://oeis.org, 2025. Sequence A007500 (Primes whose reversal in base 10 is also prime). https://oeis.org/A007500 Sloane, N. J. A. The On-Line Encyclopedia of Integer Sequences, published electronically at https://oeis.org, 2025. Sequence A002385 (Palindromic primes in base 2). https://oeis.org/A117697 Sloane, N. J. A. The On-Line Encyclopedia of Integer Sequences, published electronically at https://oeis.org, 2025. Sequence A007500 (Palindromes only using 0 and 1). https://oeis.org/A057148