A new method for finding amicable pairs:
Abstract: "Let [sigma](x) denote the sum of all divisors of the (positive) integer x. An amicable pair is a pair of integers (m,n) with m < n such that [sigma](m) = [sigma](n) = m + n. The smallest amicable pair is (220,284). A new method for finding amicable pairs is presented, based on the...
Gespeichert in:
| 1. Verfasser: | |
|---|---|
| Format: | Buch |
| Sprache: | English |
| Veröffentlicht: |
Amsterdam
1995
|
| Schriftenreihe: | Centrum voor Wiskunde en Informatica <Amsterdam> / Afdeling Numerieke Wiskunde: Report NM
1995,12 |
| Schlagworte: | |
| Zusammenfassung: | Abstract: "Let [sigma](x) denote the sum of all divisors of the (positive) integer x. An amicable pair is a pair of integers (m,n) with m < n such that [sigma](m) = [sigma](n) = m + n. The smallest amicable pair is (220,284). A new method for finding amicable pairs is presented, based on the following observation of Erdős: For given s, let x₁, x₂,... be solutions of the equation [sigma](x) = s, then any pair (x[subscript i], x[subscript j]) for which x[subscript i] + x[subscript j] = s is amicable. The problem here is to find numbers s for which the equation [sigma](x) = s has many solutions. From inspection of tables of known amicable pairs and their pair sums one learns that certain smooth numbers s (i.e., numbers with only small prime divisors) are good candidates. With the help of a precomputed table of [sigma](p[superscript e])-values, many solutions of the equation [sigma](x) = s were found by checking divisibility of s by the tabled [sigma]-values in a recursive way. In the set of solutions found, pairs were traced which sum up to s. From 1850 smooth numbers s satisfying 4 x 10¹¹ < s < 10¹² we found 116 new amicable pairs with this algorithm. After the submission of this paper to the Vancouver Conference Mathematics of Computation 1943-1993, the computations have been extended and yielded many more new amicable pairs. In particular, the first quadruple of amicable pairs with the same pair sum (namely 16!) was found. A list is given of 587 amicable pairs with smaller member [sic] between 2.01 x 10¹¹ and 10¹², of which 565 pairs seem to be new." |
| Beschreibung: | 20 S. |
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| 100 | 1 | |a Riele, Herman J. te |e Verfasser |4 aut | |
| 245 | 1 | 0 | |a A new method for finding amicable pairs |c H. J. J. te Riele |
| 264 | 1 | |a Amsterdam |c 1995 | |
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| 490 | 1 | |a Centrum voor Wiskunde en Informatica <Amsterdam> / Afdeling Numerieke Wiskunde: Report NM |v 1995,12 | |
| 520 | 3 | |a Abstract: "Let [sigma](x) denote the sum of all divisors of the (positive) integer x. An amicable pair is a pair of integers (m,n) with m < n such that [sigma](m) = [sigma](n) = m + n. The smallest amicable pair is (220,284). A new method for finding amicable pairs is presented, based on the following observation of Erdős: For given s, let x₁, x₂,... be solutions of the equation [sigma](x) = s, then any pair (x[subscript i], x[subscript j]) for which x[subscript i] + x[subscript j] = s is amicable. The problem here is to find numbers s for which the equation [sigma](x) = s has many solutions. From inspection of tables of known amicable pairs and their pair sums one learns that certain smooth numbers s (i.e., numbers with only small prime divisors) are good candidates. With the help of a precomputed table of [sigma](p[superscript e])-values, many solutions of the equation [sigma](x) = s were found by checking divisibility of s by the tabled [sigma]-values in a recursive way. In the set of solutions found, pairs were traced which sum up to s. From 1850 smooth numbers s satisfying 4 x 10¹¹ < s < 10¹² we found 116 new amicable pairs with this algorithm. After the submission of this paper to the Vancouver Conference Mathematics of Computation 1943-1993, the computations have been extended and yielded many more new amicable pairs. In particular, the first quadruple of amicable pairs with the same pair sum (namely 16!) was found. A list is given of 587 amicable pairs with smaller member [sic] between 2.01 x 10¹¹ and 10¹², of which 565 pairs seem to be new." | |
| 650 | 4 | |a Divisor theory | |
| 650 | 4 | |a Search theory | |
| 810 | 2 | |a Afdeling Numerieke Wiskunde: Report NM |t Centrum voor Wiskunde en Informatica <Amsterdam> |v 1995,12 |w (DE-604)BV010177152 |9 1995,12 | |
| 999 | |a oai:aleph.bib-bvb.de:BVB01-007407140 | ||
Datensatz im Suchindex
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| author | Riele, Herman J. te |
| author_facet | Riele, Herman J. te |
| author_role | aut |
| author_sort | Riele, Herman J. te |
| author_variant | h j t r hjt hjtr |
| building | Verbundindex |
| bvnumber | BV011059905 |
| ctrlnum | (OCoLC)35448379 (DE-599)BVBBV011059905 |
| format | Book |
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| id | DE-604.BV011059905 |
| illustrated | Not Illustrated |
| indexdate | 2024-07-09T18:03:19Z |
| institution | BVB |
| language | English |
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| oclc_num | 35448379 |
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| owner_facet | DE-91G DE-BY-TUM |
| physical | 20 S. |
| publishDate | 1995 |
| publishDateSearch | 1995 |
| publishDateSort | 1995 |
| record_format | marc |
| series2 | Centrum voor Wiskunde en Informatica <Amsterdam> / Afdeling Numerieke Wiskunde: Report NM |
| spelling | Riele, Herman J. te Verfasser aut A new method for finding amicable pairs H. J. J. te Riele Amsterdam 1995 20 S. txt rdacontent n rdamedia nc rdacarrier Centrum voor Wiskunde en Informatica <Amsterdam> / Afdeling Numerieke Wiskunde: Report NM 1995,12 Abstract: "Let [sigma](x) denote the sum of all divisors of the (positive) integer x. An amicable pair is a pair of integers (m,n) with m < n such that [sigma](m) = [sigma](n) = m + n. The smallest amicable pair is (220,284). A new method for finding amicable pairs is presented, based on the following observation of Erdős: For given s, let x₁, x₂,... be solutions of the equation [sigma](x) = s, then any pair (x[subscript i], x[subscript j]) for which x[subscript i] + x[subscript j] = s is amicable. The problem here is to find numbers s for which the equation [sigma](x) = s has many solutions. From inspection of tables of known amicable pairs and their pair sums one learns that certain smooth numbers s (i.e., numbers with only small prime divisors) are good candidates. With the help of a precomputed table of [sigma](p[superscript e])-values, many solutions of the equation [sigma](x) = s were found by checking divisibility of s by the tabled [sigma]-values in a recursive way. In the set of solutions found, pairs were traced which sum up to s. From 1850 smooth numbers s satisfying 4 x 10¹¹ < s < 10¹² we found 116 new amicable pairs with this algorithm. After the submission of this paper to the Vancouver Conference Mathematics of Computation 1943-1993, the computations have been extended and yielded many more new amicable pairs. In particular, the first quadruple of amicable pairs with the same pair sum (namely 16!) was found. A list is given of 587 amicable pairs with smaller member [sic] between 2.01 x 10¹¹ and 10¹², of which 565 pairs seem to be new." Divisor theory Search theory Afdeling Numerieke Wiskunde: Report NM Centrum voor Wiskunde en Informatica <Amsterdam> 1995,12 (DE-604)BV010177152 1995,12 |
| spellingShingle | Riele, Herman J. te A new method for finding amicable pairs Divisor theory Search theory |
| title | A new method for finding amicable pairs |
| title_auth | A new method for finding amicable pairs |
| title_exact_search | A new method for finding amicable pairs |
| title_full | A new method for finding amicable pairs H. J. J. te Riele |
| title_fullStr | A new method for finding amicable pairs H. J. J. te Riele |
| title_full_unstemmed | A new method for finding amicable pairs H. J. J. te Riele |
| title_short | A new method for finding amicable pairs |
| title_sort | a new method for finding amicable pairs |
| topic | Divisor theory Search theory |
| topic_facet | Divisor theory Search theory |
| volume_link | (DE-604)BV010177152 |
| work_keys_str_mv | AT rielehermanjte anewmethodforfindingamicablepairs |