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On 2025 and Other Torn Numbers.

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Title: On 2025 and Other Torn Numbers.
Authors: Javaheri, Mohammad (AUTHOR) mjavaheri@siena.edu
Source: American Mathematical Monthly. May2026, Vol. 133 Issue 5, p478-484. 7p.
Subject Terms: Prime factors (Mathematics), Number theory
Abstract: 2025 is an example of a torn number, a perfect square that is a fixed point of the following split-add-square procedure: the number is torn in half (as 20 and 25), then the two parts are added together ( 20 + 25 = 45 ), and finally the result is squared ( 45 2 = 2025 ). We show that there exist at least 2 μ (k) − 1 2k-digit torn numbers, k ≥ 1 , where μ (k) is the number of prime factors of 10 k − 1 . Moreover, we show that the split-add-square procedure has infinitely many periodic points of any arbitrary period. [ABSTRACT FROM AUTHOR]
Copyright of American Mathematical Monthly is the property of Taylor & Francis Ltd and its content may not be copied or emailed to multiple sites without the copyright holder's express written permission. Additionally, content may not be used with any artificial intelligence tools or machine learning technologies. However, users may print, download, or email articles for individual use. This abstract may be abridged. No warranty is given about the accuracy of the copy. Users should refer to the original published version of the material for the full abstract. (Copyright applies to all Abstracts.)
Database: Education Research Complete
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DbLabel: Education Research Complete
An: 193689639
AccessLevel: 6
PubType: Academic Journal
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  Data: 2025 is an example of a torn number, a perfect square that is a fixed point of the following split-add-square procedure: the number is torn in half (as 20 and 25), then the two parts are added together ( 20 + 25 = 45 ), and finally the result is squared ( 45 2 = 2025 ). We show that there exist at least 2 μ (k) − 1 2k-digit torn numbers, k ≥ 1 , where μ (k) is the number of prime factors of 10 k − 1 . Moreover, we show that the split-add-square procedure has infinitely many periodic points of any arbitrary period. [ABSTRACT FROM AUTHOR]
– Name: AbstractSuppliedCopyright
  Label:
  Group: Ab
  Data: <i>Copyright of American Mathematical Monthly is the property of Taylor & Francis Ltd and its content may not be copied or emailed to multiple sites without the copyright holder's express written permission. Additionally, content may not be used with any artificial intelligence tools or machine learning technologies. However, users may print, download, or email articles for individual use. This abstract may be abridged. No warranty is given about the accuracy of the copy. Users should refer to the original published version of the material for the full abstract.</i> (Copyright applies to all Abstracts.)
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        Value: 10.1080/00029890.2025.2561491
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        Text: English
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      – SubjectFull: Number theory
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              Text: May2026
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