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Journal articles on the topic 'Perfect numbers'

Author: Grafiati

Published: 4 June 2021

Last updated: 25 July 2025

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1

Ausubel, Ramona. "Perfect Numbers." Ploughshares 50, no. 2 (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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2

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

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3

Holdener, Judy, and Emily Rachfal. "Perfect and Deficient Perfect Numbers." American Mathematical Monthly 126, no. 6 (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 (2012): 3037–46. http://dx.doi.org/10.1016/j.jnt.2012.06.008.

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6

Panda, G. K., and Ravi Kumar Davala. "Perfect Balancing Numbers." Fibonacci Quarterly 53, no. 3 (2015): 261–64. http://dx.doi.org/10.1080/00150517.2015.12428268.

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7

Heath-Brown, D. R. "Odd perfect numbers." Mathematical Proceedings of the Cambridge Philosophical Society 115, no. 2 (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 how

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8

Klurman, Oleksiy. "Radical of perfect numbers and perfect numbers among polynomial values." International Journal of Number Theory 12, no. 03 (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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9

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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10

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 (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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11

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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12

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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13

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

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14

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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15

Hagis, Peter. "Odd Nonunitary Perfect Numbers." Fibonacci Quarterly 28, no. 1 (1990): 11–15. http://dx.doi.org/10.1080/00150517.1990.12429514.

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16

CHEN, FENG-JUAN, and YONG-GAO CHEN. "ON ODD PERFECT NUMBERS." Bulletin of the Australian Mathematical Society 86, no. 3 (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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17

TANG, MIN, and MIN FENG. "ON DEFICIENT-PERFECT NUMBERS." Bulletin of the Australian Mathematical Society 90, no. 2 (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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18

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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19

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

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20

KNOPFMACHER, ARNOLD, and FLORIAN LUCA. "ON PRIME-PERFECT NUMBERS." International Journal of Number Theory 07, no. 07 (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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21

CHEN, SHI-CHAO, and HAO LUO. "ODD MULTIPERFECT NUMBERS." Bulletin of the Australian Mathematical Society 88, no. 1 (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

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22

Segura, J. J., and S. Ortega. "All Known Perfect Numbers other than 6 Satisfy N=4+6n." International Journal Of Mathematics And Computer Research 12, no. 03 (2024): 4103–6. https://doi.org/10.5281/zenodo.10846916.

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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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23

Bezuszka, Stanley J., and Margaret J. Kenney. "Even Perfect Numbers: (Update)2." Mathematics Teacher 90, no. 8 (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 determ

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24

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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25

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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26

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

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27

Kalita, Bichitra. "SEARCHING OF INFINITE PERFECT NUMBERS." Advances in Mathematics: Scientific Journal 14, no. 2 (2025): 247–66. https://doi.org/10.37418/amsj.14.2.8.

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It is well known that only 51 perfect numbers have been discovered in number system. The present study gives an overview of existence of infinite perfect numbers with the help of a theoretical investigation. In this the investigation, it is found that all infinite perfect numbers belong to a special type of set. Some properties, propositions and important results have also been discussed.

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28

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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29

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

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30

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

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31

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

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32

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

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33

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

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34

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

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35

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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36

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

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37

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

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38

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 (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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39

YUAN, PINGZHI. "AN UPPER BOUND FOR THE NUMBER OF ODD MULTIPERFECT NUMBERS." Bulletin of the Australian Mathematical Society 89, no. 1 (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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40

Uma, Dixit. "On Bounds of Non-Deficient Numbers." Indian Journal of Science and Technology 15, no. 35 (2022): 1683–90. https://doi.org/10.17485/IJST/v15i35.946.

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Abstract <strong>Objectives:</strong> To improve the upper bounds of a quasi perfect number and give an important result on its divisibility with primes.<strong> Methods:</strong> A positive integer n is quasi perfect if s (n) >2n + 1, where s (n) denotes the sum of the positive divisors of n. However, the existence of a quasi perfect number, which is a Non-Deficient number, is still an open problem. We use R(n), the sum of the reciprocals of distinct primes dividing the quasi perfect number, to derive lemmas and improve the bounds obtained by earlier authors.<strong> Fi

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41

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 (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

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42

Cai, Tianxin, Deyi Chen, and Yong Zhang. "Perfect numbers and Fibonacci primes (I)." International Journal of Number Theory 11, no. 01 (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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43

Bravo, Jhon J., and Florian Luca. "Perfect Pell and Pell–Lucas numbers." Studia Scientiarum Mathematicarum Hungarica 56, no. 4 (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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44

Sándor, József. "On certain rational perfect numbers, II." Notes on Number Theory and Discrete Mathematics 28, no. 3 (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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45

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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46

Aursukaree, Saralee, and Prapanpong Pongsriiam. "On Exactly 3-Deficient-Perfect Numbers." Fibonacci Quarterly 59, no. 1 (2021): 33–46. http://dx.doi.org/10.1080/00150517.2021.12427539.

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47

Yabuta, Minoru. "Perfect Squares in the Lucas Numbers." Fibonacci Quarterly 40, no. 5 (2002): 460–66. http://dx.doi.org/10.1080/00150517.2002.12428625.

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48

Dris, Jose Arnaldo B., and Florian Luca. "A Note on Odd Perfect Numbers." Fibonacci Quarterly 54, no. 4 (2016): 291–95. http://dx.doi.org/10.1080/00150517.2016.12427796.

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49

CHEN, SHI-CHAO, and HAO LUO. "BOUNDS FOR ODD k-PERFECT NUMBERS." Bulletin of the Australian Mathematical Society 84, no. 3 (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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50

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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