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Journal articles on the topic 'Perfect numbers'

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Published: 4 June 2021
Last updated: 29 July 2024

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1

Hassler, Uwe. "Perfect Numbers." Euleriana 3, no. 2 (August 22, 2023): 176–85. http://dx.doi.org/10.56031/2693-9908.1052.

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2

Ausubel, Ramona. "Perfect Numbers." Ploughshares 50, no. 2 (June 2024): 32–46. http://dx.doi.org/10.1353/plo.2024.a932313.

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Abstract: The Summer 2024 Issue. Ploughshares is an award-winning journal of new writing. Since 1971, Ploughshares has discovered and cultivated the freshest voices in contemporary American literature, and now provides readers with thoughtful and entertaining literature in a variety of formats. Find out why the New York Times named Ploughshares “the Triton among minnows.” The Summer 2024 Issue, guest-edited by Rebecca Makkai, features prose by Dur e Aziz Amna, Ramona Ausubel, Peter Mountford, Khaddafina Mbabazi, DK Nnuro, and more.
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3

Holdener, Judy, and Emily Rachfal. "Perfect and Deficient Perfect Numbers." American Mathematical Monthly 126, no. 6 (May 29, 2019): 541–46. http://dx.doi.org/10.1080/00029890.2019.1584515.

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4

Fu, Ruiqin, Hai Yang, and Jing Wu. "The Perfect Numbers of Pell Number." Journal of Physics: Conference Series 1237 (June 2019): 022041. http://dx.doi.org/10.1088/1742-6596/1237/2/022041.

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5

Pollack, Paul, and Vladimir Shevelev. "On perfect and near-perfect numbers." Journal of Number Theory 132, no. 12 (December 2012): 3037–46. http://dx.doi.org/10.1016/j.jnt.2012.06.008.

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6

Heath-Brown, D. R. "Odd perfect numbers." Mathematical Proceedings of the Cambridge Philosophical Society 115, no. 2 (March 1994): 191–96. http://dx.doi.org/10.1017/s0305004100072030.

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It is not known whether or not odd perfect numbers can exist. However it is known that there is no such number below 10300 (see Brent[1]). Moreover it has been proved by Hagis[4]and Chein[2] independently that an odd perfect number must have at least 8 prime factors. In fact results of this latter type can in priniciple be obtained solely by calculation, in view of the result of Pomerance[6] who showed that if N is an odd perfect number with at most k prime factors, thenPomerance's work was preceded by a theorem of Dickson[3]showing that there can be only a finite number of such N. Clearly however the above bound is vastly too large to be of any practical use. The principal object of the present paper is to sharpen the estimate (1). Indeed we shall handle odd ‘multiply perfect’ numbers in general, as did Kanold[5], who extended Dickson's work, and Pomerance. Our result is the following.
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7

Klurman, Oleksiy. "Radical of perfect numbers and perfect numbers among polynomial values." International Journal of Number Theory 12, no. 03 (March 23, 2016): 585–91. http://dx.doi.org/10.1142/s1793042116500378.

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It is conjectured that [Formula: see text] for any perfect number [Formula: see text]. We prove that [Formula: see text] improving the previous bound of Luca and Pomerance as well as Acquaah and Konyagin. As a consequence, we prove that assuming the [Formula: see text]-conjecture, any integer polynomial of degree [Formula: see text] without repeated factors can take only finitely many perfect values. We also show that the latter holds unconditionally for even perfect numbers.
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8

Tang, Min, Xiao-Zhi Ren, and Meng Li. "On near-perfect and deficient-perfect numbers." Colloquium Mathematicum 133, no. 2 (2013): 221–26. http://dx.doi.org/10.4064/cm133-2-8.

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9

J. J., Segura, and Ortega S. "All KnownPerfect Numbers other than 6 Satisfy N=4+6n." international journal of mathematics and computer research 12, no. 03 (March 23, 2024): 4103–6. http://dx.doi.org/10.47191/ijmcr/v12i3.04.

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For all 51 known perfect numbers ranging from (p=2 to p= 82589933) and with the only exception of N=6, all perfect numbers belong to the group of natural numbers formed by N=4+6n. If this observation can be proven valid for all existing even perfect numbers, that would automatically exclude 2/3 of all even numbers out of the possibility of being perfect. If this can be proven a necessary condition for all perfect numbers, then it would rule out the possibility of having any odd perfect numbers.
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10

Jiang, Xing-Wang. "On even perfect numbers." Colloquium Mathematicum 154, no. 1 (2018): 131–36. http://dx.doi.org/10.4064/cm7374-11-2017.

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11

Cowles, John, and Ruben Gamboa. "Perfect Numbers in ACL2." Electronic Proceedings in Theoretical Computer Science 192 (September 18, 2015): 53–59. http://dx.doi.org/10.4204/eptcs.192.5.

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12

Bencze, Mihály. "About k-Perfect Numbers." Analele Universitatii "Ovidius" Constanta - Seria Matematica 22, no. 1 (December 10, 2014): 45–50. http://dx.doi.org/10.2478/auom-2014-0005.

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13

Cohen, Peter, Katherine Cordwell, Alyssa Epstein, Chung-Hang Kwan, Adam Lott, and Steven J. Miller. "On near-perfect numbers." Acta Arithmetica 194, no. 4 (2020): 341–66. http://dx.doi.org/10.4064/aa180821-11-10.

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14

CHEN, FENG-JUAN, and YONG-GAO CHEN. "ON ODD PERFECT NUMBERS." Bulletin of the Australian Mathematical Society 86, no. 3 (February 16, 2012): 510–14. http://dx.doi.org/10.1017/s0004972712000032.

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AbstractLet q be an odd prime. In this paper, we prove that if N is an odd perfect number with qα∥N then σ(N/qα)/qα≠p,p2,p3,p4,p1p2,p21p2, where p,p1, p2 are primes and p1≠p2. This improves a result of Dris and Luca [‘A note on odd perfect numbers’, arXiv:1103.1437v3 [math.NT]]: σ(N/qα)/qα≠1,2,3,4,5. Furthermore, we prove that for K≥1 , if N is an odd perfect number with qα ∥N and σ(N/qα)/qα ≤K, then N≤4K8.
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15

TANG, MIN, and MIN FENG. "ON DEFICIENT-PERFECT NUMBERS." Bulletin of the Australian Mathematical Society 90, no. 2 (May 23, 2014): 186–94. http://dx.doi.org/10.1017/s0004972714000082.

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AbstractFor a positive integer $\def \xmlpi #1{}\def \mathsfbi #1{\boldsymbol {\mathsf {#1}}}\let \le =\leqslant \let \leq =\leqslant \let \ge =\geqslant \let \geq =\geqslant \def \Pr {\mathit {Pr}}\def \Fr {\mathit {Fr}}\def \Rey {\mathit {Re}}n$, let $\sigma (n)$ denote the sum of the positive divisors of $n$. Let $d$ be a proper divisor of $n$. We call $n$ a deficient-perfect number if $\sigma (n) = 2n - d$. In this paper, we show that there are no odd deficient-perfect numbers with three distinct prime divisors.
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16

Finch, Carrie, and Lenny Jones. "Perfect power Riesel numbers." Journal of Number Theory 150 (May 2015): 41–46. http://dx.doi.org/10.1016/j.jnt.2014.11.004.

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17

Dittmer, Samuel J. "Spoof odd perfect numbers." Mathematics of Computation 83, no. 289 (October 25, 2013): 2575–82. http://dx.doi.org/10.1090/s0025-5718-2013-02793-7.

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18

KNOPFMACHER, ARNOLD, and FLORIAN LUCA. "ON PRIME-PERFECT NUMBERS." International Journal of Number Theory 07, no. 07 (November 2011): 1705–16. http://dx.doi.org/10.1142/s1793042111004447.

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We prove that the Diophantine equation [Formula: see text] has only finitely many positive integer solutions k, p1, …, pk, r1, …, rk, where p1, …, pk are distinct primes. If a positive integer n has prime factorization [Formula: see text], then [Formula: see text] represents the number of ordered factorizations of n into prime parts. Hence, solutions to the above Diophantine equation are designated as prime-perfect numbers.
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19

CHEN, SHI-CHAO, and HAO LUO. "ODD MULTIPERFECT NUMBERS." Bulletin of the Australian Mathematical Society 88, no. 1 (November 6, 2012): 56–63. http://dx.doi.org/10.1017/s0004972712000858.

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AbstractA natural number $n$ is called multiperfect or $k$-perfect for integer $k\ge 2$ if $\sigma (n)=kn$, where $\sigma (n)$ is the sum of the positive divisors of $n$. In this paper, we establish a theorem on odd multiperfect numbers analogous to Euler’s theorem on odd perfect numbers. We describe the divisibility of the Euler part of odd multiperfect numbers and characterise the forms of odd perfect numbers $n=\pi ^\alpha M^2$ such that $\pi \equiv \alpha ~({\rm mod}~8)$, where $\pi ^\alpha $ is the Euler factor of $n$. We also present some examples to show the nonexistence of odd perfect numbers of certain forms.
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20

Bezuszka, Stanley J., and Margaret J. Kenney. "Even Perfect Numbers: (Update)2." Mathematics Teacher 90, no. 8 (November 1997): 628–33. http://dx.doi.org/10.5951/mt.90.8.0628.

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Perfect Numbers may appear perfectly useless; however, they do have redeeming features. Specifically, the pursuit of perfect numbers leads us to examine the history of mathematics very closely to locate information about the progression of mathematicians who have discovered and worked with them. Students assigned to produce a report on perfect numbers and their properties will uncover some fascinating episodes. Further, perfect numbers are part of the frontier of the technological age. They are woven into the mystique of the supercomputer. Anyone with an interest in computing can try to determine how the computer helps in the search for perfect numbers and what algorithms are used for this purpose.
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21

Nelsen, Roger B. "Proof Without Words: Perfect Numbers and Triangular Numbers." College Mathematics Journal 47, no. 3 (May 2016): 171. http://dx.doi.org/10.4169/college.math.j.47.3.171.

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22

Ibro, Vait, and Eugen Ljajko. "Prime, perfect and friendly numbers." Zbornik radova Uciteljskog fakulteta Prizren-Leposavic, no. 12 (2018): 29–39. http://dx.doi.org/10.5937/zrufpl1812029i.

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23

Das, Bhabesh, and Helen K. Saikia. "On Near 3−Perfect Numbers." Sohag Journal of Mathematics 4, no. 1 (January 1, 2017): 1–5. http://dx.doi.org/10.18576/sjm/040101.

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24

Beldon, Tom, and Tony Gardiner. "Triangular Numbers and Perfect Squares." Mathematical Gazette 86, no. 507 (November 2002): 423. http://dx.doi.org/10.2307/3621134.

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25

Sándor, József. "On certain rational perfect numbers." Notes on Number Theory and Discrete Mathematics 28, no. 2 (May 12, 2022): 281–85. http://dx.doi.org/10.7546/nntdm.2022.28.2.281-285.

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26

Flowers, Joe. "Some Characterizations of Perfect Numbers." Missouri Journal of Mathematical Sciences 7, no. 3 (October 1995): 104–15. http://dx.doi.org/10.35834/1995/0703104.

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27

Popov, Michael A. "On Plato's periodic perfect numbers." Bulletin des Sciences Mathématiques 123, no. 1 (January 1999): 29–31. http://dx.doi.org/10.1016/s0007-4497(99)80011-6.

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28

Kurokawa, Nobushige, and Wakayama Masato. "Zeta functions ofq-perfect numbers." Rendiconti del Circolo Matematico di Palermo 53, no. 3 (October 2004): 381–89. http://dx.doi.org/10.1007/bf02875730.

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29

De Medts, Tom, and Attila Maróti. "Perfect numbers and finite groups." Rendiconti del Seminario Matematico della Università di Padova 129 (2013): 17–33. http://dx.doi.org/10.4171/rsmup/129-2.

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30

Luca, Florian. "Perfect fibonacci and lucas numbers." Rendiconti del Circolo Matematico di Palermo 49, no. 2 (May 2000): 313–18. http://dx.doi.org/10.1007/bf02904236.

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31

Dunham, William. "Odd Perfect Numbers: A Triptych." Mathematical Intelligencer 42, no. 1 (August 22, 2019): 42–46. http://dx.doi.org/10.1007/s00283-019-09915-6.

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32

Bhutani, Kiran R., and Alexander B. Levin. "Graceful numbers." International Journal of Mathematics and Mathematical Sciences 29, no. 8 (2002): 495–99. http://dx.doi.org/10.1155/s0161171202007615.

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We construct a labeled graphD(n)that reflects the structure of divisors of a given natural numbern. We define the concept of graceful numbers in terms of this associated graph and find the general form of such a number. As a consequence, we determine which graceful numbers are perfect.
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33

Gandhi, K. Raja Rama. "Note on Perfect Numbers and their Existence." Bulletin of Mathematical Sciences and Applications 3 (February 2013): 15–19. http://dx.doi.org/10.18052/www.scipress.com/bmsa.3.15.

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This paper will address the interesting results on perfect numbers. As we know that,perfect number ends with 6 or 8 and perfect numbers had some special relation with primes. Hereone can understand that the reasons of relation with primes and existence of odd perfect numbers. If exists, the structures of odd perfect numbers in modulo.
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34

YUAN, PINGZHI. "AN UPPER BOUND FOR THE NUMBER OF ODD MULTIPERFECT NUMBERS." Bulletin of the Australian Mathematical Society 89, no. 1 (January 28, 2013): 1–4. http://dx.doi.org/10.1017/s000497271200113x.

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AbstractA natural number $n$ is called $k$-perfect if $\sigma (n)= kn$. In this paper, we show that for any integers $r\geq 2$ and $k\geq 2$, the number of odd $k$-perfect numbers $n$ with $\omega (n)\leq r$ is bounded by $\left({\lfloor {4}^{r} { \mathop{ \log } \nolimits }_{3} 2\rfloor + r\atop r} \right){ \mathop{ \sum } \nolimits }_{i= 1}^{r} \left({\lfloor kr/ 2\rfloor \atop i} \right)$, which is less than ${4}^{{r}^{2} } $ when $r$ is large enough.
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35

REN, XIAO-ZHI, and YONG-GAO CHEN. "ON NEAR-PERFECT NUMBERS WITH TWO DISTINCT PRIME FACTORS." Bulletin of the Australian Mathematical Society 88, no. 3 (March 11, 2013): 520–24. http://dx.doi.org/10.1017/s0004972713000178.

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AbstractRecently, Pollack and Shevelev [‘On perfect and near-perfect numbers’, J. Number Theory 132 (2012), 3037–3046] introduced the concept of near-perfect numbers. A positive integer $n$ is called near-perfect if it is the sum of all but one of its proper divisors. In this paper, we determine all near-perfect numbers with two distinct prime factors.
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36

Cai, Tianxin, Deyi Chen, and Yong Zhang. "Perfect numbers and Fibonacci primes (I)." International Journal of Number Theory 11, no. 01 (November 24, 2014): 159–69. http://dx.doi.org/10.1142/s1793042115500098.

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In this paper, we introduce the concept of F-perfect number, which is a positive integer n such that ∑d|n,d<n d2 = 3n. We prove that all the F-perfect numbers are of the form n = F2k-1 F2k+1, where both F2k-1 and F2k+1 are Fibonacci primes. Moreover, we obtain other interesting results and raise a new conjecture on perfect numbers.
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37

Bravo, Jhon J., and Florian Luca. "Perfect Pell and Pell–Lucas numbers." Studia Scientiarum Mathematicarum Hungarica 56, no. 4 (December 2019): 381–87. http://dx.doi.org/10.1556/012.2019.56.4.1440.

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Abstract The Pell sequence is given by the recurrence Pn = 2Pn−1 + Pn−2 with initial condition P0 = 0, P1 = 1 and its associated Pell-Lucas sequence is given by the same recurrence relation but with initial condition Q0 = 2, Q1 = 2. Here we show that 6 is the only perfect number appearing in these sequences. This paper continues a previous work that searched for perfect numbers in the Fibonacci and Lucas sequences.
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38

Sándor, József. "On certain rational perfect numbers, II." Notes on Number Theory and Discrete Mathematics 28, no. 3 (August 10, 2022): 525–32. http://dx.doi.org/10.7546/nntdm.2022.28.3.525-532.

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We continue the study from [1], by studying equations of type $\psi(n) = \dfrac{k+1}{k} \cdot \ n+a,$ $a\in \{0, 1, 2, 3\},$ and $\varphi(n) = \dfrac{k-1}{k} \cdot \ n-a,$ $a\in \{0, 1, 2, 3\}$ for $k > 1,$ where $\psi(n)$ and $\varphi(n)$ denote the Dedekind, respectively Euler's, arithmetical functions.
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39

Kustin, Andrew R. "Perfect modules with Betti numbers (2,6,5,1)." Journal of Algebra 600 (June 2022): 71–124. http://dx.doi.org/10.1016/j.jalgebra.2022.02.005.

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40

CHEN, SHI-CHAO, and HAO LUO. "BOUNDS FOR ODD k-PERFECT NUMBERS." Bulletin of the Australian Mathematical Society 84, no. 3 (July 21, 2011): 475–80. http://dx.doi.org/10.1017/s0004972711002462.

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AbstractLet k≥2 be an integer. A natural number n is called k-perfect if σ(n)=kn. For any integer r≥1, we prove that the number of odd k-perfect numbers with at most r distinct prime factors is bounded by (k−1)4r3.
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41

Becher, Verónica, and Olivier Carton. "Normal numbers and nested perfect necklaces." Journal of Complexity 54 (October 2019): 101403. http://dx.doi.org/10.1016/j.jco.2019.03.003.

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42

Fletcher, S. Adam, Pace P. Nielsen, and Pascal Ochem. "Sieve methods for odd perfect numbers." Mathematics of Computation 81, no. 279 (September 1, 2012): 1753–76. http://dx.doi.org/10.1090/s0025-5718-2011-02576-7.

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43

Gallardo, Luis H., and Olivier Rahavandrainy. "New Congruences for Odd Perfect Numbers." Rocky Mountain Journal of Mathematics 36, no. 1 (February 2006): 225–35. http://dx.doi.org/10.1216/rmjm/1181069496.

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44

Lorenzini, Anna. "Betti numbers of perfect homogeneous ideals." Journal of Pure and Applied Algebra 60, no. 3 (October 1989): 273–88. http://dx.doi.org/10.1016/0022-4049(89)90088-1.

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45

Ndegwa, Duncan, Loyford Njagi, and Josephine Mutembei. "Application of Partitions of Odd Numbers and their Odd Sums to Prove the Nonexistence of Odd Perfect Numbers." Asian Research Journal of Mathematics 20, no. 5 (May 25, 2024): 28–37. http://dx.doi.org/10.9734/arjom/2024/v20i5800.

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Perfect numbers, which are integers equal to the sum of their proper divisors, excluding themselves, have intrigued mathematicians for centuries. While it is established that even perfect numbers can be expressed as 2p-1(2p-1), where p and 2p-1 are prime numbers (Mersenne primes), the existence of odd perfect numbers remains an unsolved problem. This study aims to prove the nonexistence of odd perfect numbers by utilizing an algorithm which demonstrates that a positive even integer can be partitioned into all pairs of odd numbers. Using this approach, it is proven that any positive odd number 2n+1 can be partitioned into all pairs of both odd and even numbers and from the set of these partitions, we show that there exist a proper subset containing all proper divisors of 2n+1. Using these results, and the facts that there exist infinitely many odd numbers and the odd sums of odd numbers is always odd, we prove the nonexistence of odd perfect numbers contributing to the conjecture that they do not exist.
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46

Sumicad, Rulthan P. "On the Picture-Perfect Number." Journal of Mathematics and Statistics Studies 4, no. 4 (December 4, 2023): 106–11. http://dx.doi.org/10.32996/jmss.2023.4.4.11.

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This is a seminar paper on the article picture perfect numbers by Joseph L. Pe that was published in the journal mathematical spectrum in 2008. This paper begins with a discussion of the definition of the more familiar concept of perfect numbers, then proceeds to a discussion of the picture-perfect numbers as defined by Joseph L. Pe, and winds up with a discussion on how to obtain a picture-perfect number using the Andersen's Theorem. This paper also includes proof of the Andersen's Theorem, as well as that of the Andersen's Lemma, which are both attributed to Jens Kruce Andersen.
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47

Asadulla, Syed. "Thirty-nine perfect numbers and their divisors." International Journal of Mathematics and Mathematical Sciences 9, no. 1 (1986): 205–6. http://dx.doi.org/10.1155/s016117128600025x.

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The following results concerning even perfect numbers and their divisors are proved: (1) A positive integernof the form2p−1(2p−1), where2p−1is prime, is a perfect number; (2) every even perfect number is a triangular number; (3)τ(n)=2p, whereτ(n)is the number of positive divisors ofn; (4) the product of the positive divisors ofnisnp; and (5) the sum of the reciprocals of the positive divisors ofnis2. Values ofpfor which 30 even perfect numbers have been found so far are also given.
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48

Goto, Takeshi. "Upper Bounds for Unitary Perfect Numbers and Unitary Harmonic Numbers." Rocky Mountain Journal of Mathematics 37, no. 5 (October 2007): 1557–76. http://dx.doi.org/10.1216/rmjm/1194275935.

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49

Jakimczuk, Rafael. "Divisors of numbers with k prime factors and perfect numbers." International Mathematical Forum 10 (2015): 339–47. http://dx.doi.org/10.12988/imf.2015.5435.

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50

Barbeau, E. J. "Numbers Differing from Consecutive Squares by Squares." Canadian Mathematical Bulletin 28, no. 3 (September 1, 1985): 337–42. http://dx.doi.org/10.4153/cmb-1985-040-9.

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AbstractIt is shown that there are infinitely many natural numbers which differ from the next four greater perfect squares by a perfect square. This follows from the determination of certain families of solutions to the diophantine equation 2(b2 + 1) = a2 + c2. However, it is essentially known that any natural number with this property cannot be 1 less than a perfect square. The question whether there exists a number differing from the next five greater squares by squares is open.
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