Don't take life too seriously: it's just a temporary situation

Indice

  1. 1. Pagina principale
  2. 2. Primi vicini a potenze di due
  3. 3. Proprietà delle cifre delle potenze di due

Per la scomposizione in fattori primi di numeri della forma 2n ± 1 v. numeri di Cunningham.

 

Erdös nel 1957 chiese per quali valori di n 2n – 7 sia primo; non fu troppo difficile trovare la prima soluzione, n = 39, mentre ci volle parecchio per le successive.

 

La tabella seguente riporta gli esponenti per i quali 2nk è primo; per quasi tutti i maggiori valori di n i primi sono solo primi probabili. Non si sa se, fissato k, vi siano infinite soluzioni o meno.

k

n

1

Primi di Mersenne

3

3, 4, 5, 6, 9, 10, 12, 14, 20, 22, 24, 29, 94, 116, 122, 150, 174, 213, 221, 233, 266, 336, 452, 545, 689, 694, 850, 1736, 2321, 3237, 3954, 5630, 6756, 8770, 10572, 14114, 14400, 16460, 16680, 20757, 26350, 30041, 34452* (Paul Underwood, 2002), 36552* (Paul Underwood, 2002), 42689* (Paul Underwood, 2002), 44629* (Paul Underwood, 2002), 50474* (Paul Underwood, 2002), 66422* (Paul Underwood, 2002), 69337* (Paul Underwood, 2002), 116926* (M. Frind e Paul Underwood, 2002), 119324* (M. Frind e Paul Underwood, 2002), 123297* (Gary Barnes, 2008), 189110* (M. Frind e Paul Underwood, 2002), 241004* (M. Frind e Paul Underwood, 2002), 247165* (M. Frind e Paul Underwood, 2002), 284133* (M. Frind e Paul Underwood, 2002), 354946* (M. Frind e Paul Underwood, 2002), 394034* (M. Frind e Paul Underwood, 2002), 702194* (Paul Bourdelais, 2012), 750740* (Paul Bourdelais, 2012)

5

3, 4, 6, 8, 10, 12, 18, 20, 26, 32, 36, 56, 66, 118, 130, 150, 166, 206, 226, 550, 706, 810, 1136, 1228, 1818, 2368, 2400, 3128, 4532, 5112, 8492, 16028, 16386, 17392, 18582, 21986, 24292, 27618, 30918, 32762, 48212* (Paul Underwood, 2001), 120440* (Henri Lifchitz, 2005), 183632* (Henri Lifchitz, 2005), 316140* (Henri Lifchitz, 2006), 364982* (Henri Lifchitz, 2006), 414032* (Henri Lifchitz, 2006)

7

39, 715, 1983, 2319, 2499, 3775, 12819, 63583* (Henri Lifchitz, 2005), 121555* (Gary Barnes, 2008), 121839* (Gary Barnes, 2008)

9

4, 5, 9, 11, 17, 21, 33, 125, 141, 243, 251, 285, 321, 537, 563, 699, 729, 2841, 3365, 8451, 8577, 9699, 9725, 21011, 22689, 33921* (Paul Underwood, 2001), 51761* (Paul Underwood, 2001)

11

4, 6, 10, 18, 42, 78, 94, 114, 190, 322, 546, 3894, 10318, 11650, 12474, 20994, 61810* (Henri Lifchitz, 2002), 103882* (Henri Lifchitz, 2002), 296010* (Lelio R. Paula, 2012)

13

4, 5, 9, 13, 17, 57, 105, 137, 3217, 3229, 4233, 6097, 8757, 11457, 12073, 15425, 40117* (Henri Lifchitz, 2005), 45357* (Henri Lifchitz, 2005), 334809* (Lelio R. Paula, 2011)

15

5, 7, 8, 10, 14, 16, 23, 76, 95, 100, 158, 196, 235, 338, 620, 1646, 1850, 1891, 3833, 4394, 5194, 6017, 6070, 8824, 9955, 11399, 12250, 28723, 32057, 45494* (Henri Lifchitz, 2005), 137359* (Gary Barnes, 2008), 139627* (Gary Barnes, 2008), 160654* (Lelio R. Paula, 2010), 178819* (Lelio R. Paula, 2011), 183284* (Lelio R. Paula, 2011), 276391* (Lelio R. Paula, 2012), 283466* (Lelio R. Paula, 2012)

17

6, 8, 12, 16, 18, 20, 22, 24, 32, 36, 42, 44, 96, 104, 152, 174, 198, 336, 414, 444, 468, 488, 664, 808, 848, 3632, 4062, 5586, 5904, 6348, 8628, 9224, 9916, 13136, 15966, 17120, 17568, 17652, 20560, 31572, 33644* (Paul Underwood, 2001), 104098* (Gary Barnes, 2008), 115842* (Gary Barnes, 2008), 130572* (Gary Barnes, 2008), 164110* (Lelio R. Paula, 2010), 189414* (Lelio R. Paula, 2011), 205110* (Lelio R. Paula, 2011)

19

5, 7, 11, 15, 19, 21, 31, 39, 67, 69, 85, 157, 171, 191, 255, 291, 379, 3669, 4551, 9531, 13119, 14211, 20647, 233965* (Lelio R. Paula, 2012), 337267* (Lelio R. Paula, 2011)

21

5, 6, 7, 9, 11, 13, 14, 21, 23, 41, 46, 89, 110, 389, 413, 489, 869, 1589, 1713, 2831, 10843, 11257, 16949, 24513, 39621* (Donovan Johnson, 2004), 43349* (Henri Lifchitz, 2005), 62941* (Henri Lifchitz, 2005), 96094* (Henri Lifchitz, 2005), 139237* (Lelio R. Paula, 2010), 145289* (Lelio R. Paula, 2010), 264683* (Lelio R. Paula, 2012)

 

Per molti valori di k potrebbero esserci altri primi per valori di n inferiori al massimo riportato.

 

La tabella seguente riporta gli esponenti per i quali 2n + k è primo; per quasi tutti i maggiori valori di n i primi sono solo primi probabili. Non si sa se, fissato k, vi siano infinite soluzioni o meno.

k

n

1

Primi di Fermat

3

1, 2, 3, 4, 6, 7, 12, 15, 16, 18, 28, 30, 55, 67, 84, 228, 390, 784, 1110, 1704, 2008, 2139, 2191, 2367, 2370, 4002, 4060, 4062, 4552, 5547, 8739, 17187, 17220, 17934, 20724, 22732, 25927, 31854, 33028, 35754* (Mike Oakes, 2001), 38244* (Mike Oakes, 2001), 39796* (Mike Oakes, 2001), 40347* (Mike Oakes, 2001), 55456* (Mike Oakes, 2001), 58312* (Mike Oakes, 2001), 122550* (Mike Oakes, 2001), 205962* (Donovan Johnson, 2006), 235326* (Donovan Johnson, 2006), 363120* (Donovan Johnson, 2006), 479844* (Donovan Johnson, 2006), 685578* (Paul Bourdelais, 2012)

5

1, 3, 5, 11, 47, 53, 141, 143, 191, 273, 341, 16541, 34001* (Marcin Lipinski, 2004), 34763* (Payam Samidoost, 2004), 42167* (Payam Samidoost, 2004), 193965* (Henri Lifchitz, 2002), 282203* (Lelio R. Paula, 2012)

7

2, 4, 6, 8, 10, 16, 18, 20, 28, 30, 38, 44, 78, 88, 98, 126, 160, 174, 204, 214, 588, 610, 798, 926, 1190, 1198, 1806, 1888, 2648, 3454, 3510, 3864, 3870, 8970, 12330, 13330, 39718* (Paul Underwood, 2002), 55006* (Paul Underwood, 2002), 110784* (Payam Samidoost, 2004), 172470* (Lelio R. Paula, 2012), 196434* (Lelio R. Paula, 2011), 235710* (Lelio R. Paula, 2012), 247280* (Lelio R. Paula, 2012), 268408* (Lelio R. Paula, 2012), 279320* (Lelio R. Paula, 2012), 300874* (Lelio R. Paula, 2012), 315268* (Lelio R. Paula, 2012), 566496* (Donovan Johnson, 2006)

9

1, 2, 3, 5, 6, 7, 9, 10, 18, 23, 30, 37, 47, 57, 66, 82, 95, 119, 175, 263, 295, 317, 319, 327, 670, 697, 886, 1342, 1717, 1855, 2394, 2710, 3229, 3253, 3749, 4375, 4494, 4557, 5278, 5567, 9327, 10129, 12727, 13615, 14893, 16473, 23639, 40053* (Mike Oakes, 2001), 44399* (Mike Oakes, 2001), 50335* (Mike Oakes, 2001), 80949* (Mike Oakes, 2001), 146397* (Gary Barnes, 2008), 173727* (Lelio R. Paula, 2011), 234586* (Lelio R. Paula, 2012), 294327* (Lelio R. Paula, 2012)

11

1, 3, 5, 7, 9, 15, 23, 29, 31, 55, 71, 77, 297, 573, 1301, 1555, 1661, 4937, 5579, 6191, 6847, 6959, 47093* (Henri Lifchitz, 2002), 74167* (Henri Lifchitz, 2002), 149039* (Gary Barnes, 2008),175137* (Lelio R. Paula, 2011) 210545* (Lelio R. Paula, 2011),240295* (Lelio R. Paula, 2012), 345547* (Lelio R. Paula, 2011)

13

2, 4, 8, 20, 38, 64, 80, 292, 1132, 4108

15

1, 2, 3, 4, 5, 6, 8, 10, 11, 12, 15, 16, 22, 23, 26, 30, 32, 40, 42, 46, 61, 72, 76, 155, 180, 198, 203, 310, 328, 342, 508, 510, 515, 546, 808, 1563, 2772, 3882, 3940, 4840, 7518, 11118, 11552, 11733

17

1, 13, 21, 33, 81, 129, 285, 297, 769, 3381, 4441, 7065

19

2, 6, 30, 162, 654, 714, 1370, 1662, 1722, 2810

21

1, 3, 4, 5, 7, 8, 11, 15, 16, 19, 44, 48, 51, 52, 61, 163, 196, 456, 492, 911, 997, 1616, 1631, 1647, 1803, 1899, 3112, 3584, 3956

Bibliografia

  • Bellos, Axel;  Il meraviglioso mondo dei numeri, Torino, Einaudi, 2011 -

    Trad. di Alex’s Adventures in Numberland. Dispatches from the Wonderful World of Mathematics, 2010.

  • De Koninck, Jean-Marie;  Those Fascinating Numbers, American Mathematical Society, 2009 -

    Un'inesauribile miniera di notizie sugli interi, informazioni e spunti per approfondimenti.

  • Menninger, Karl;  Number Words and Number Symbols, New York, Dover Publications Inc., 1992 -

    Ripubblicazione del testo pubblicato da MIT Press, Cambridge, 1969, trad. di Zahlwort und Ziffer: Eine Kulturgeschichte der Zahlen, Göttingen, Vandenoeck & Ruprecht Publishing Company, 1957-58. Un testo erudito sui termini e simboli usati per rappresentare i numeri.

  • Roberts, Joe;  The Lure of the Integers, The Mathematical Association of America, 1992 -

    Una miniera di informazioni sugli interi.

  • Winkler, Peter;  Mathematical Puzzles, Wellesley, A.K. Peters Ltd., 2004.

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