2015 y los números pitagóricos

PorJosé A. Alonso 8-enero-201530-noviembre-2015

Un número pitagórico es un número natural cuyo cuadrado se puede escribir como la suma de los cuadrados de dos números naturales no nulos; es decir, el número natural a es pitagórico si existen dos números naturales b y c distintos de cero tales que a² = b²+c². Por ejemplo, 5 es un número pitagórico ya que 5² = 3²+4² y también lo es 2015 ya que 2015² = 1612²+1209².

Definir la sucesión

    pitagoricos :: [Integer]

1	pitagoricos :: [Integer]

cuyos elementos son los números pitagóricos. Por ejemplo,

   ghci> take 20 pitagoricos
   [5,10,13,15,17,20,25,26,29,30,34,35,37,39,40,41,45,50,51,52]

1 2	ghci> take 20 pitagoricos [5,10,13,15,17,20,25,26,29,30,34,35,37,39,40,41,45,50,51,52]

Calcular la posición de 2015 en la sucesión.

Soluciones

import Test.QuickCheck

-- 1ª definición
-- =============

pitagoricos :: [Integer]
pitagoricos = [n | n <- [1..], esPitagorico n]

-- (esPitagorico n) se verifica si n es pitagórico. Por ejemplo,
--    esPitagorico 5  ==  True
--    esPitagorico 6  ==  False
esPitagorico :: Integer -> Bool
esPitagorico n = not (null (ternasPitagoricas n))

-- (ternasPitagoricas n) es la lista de las ternas (n,a,b) tales que 
-- n² = a²+b² con b < a < n. Por ejemplo,
--    ghci> ternasPitagoricas 5
--    [(5,4,3)]
--    ghci> ternasPitagoricas 2015
--    [(2015,1612,1209),(2015,1736,1023),(2015,1860,775),(2015,1953,496)]
ternasPitagoricas :: Integer -> [(Integer,Integer,Integer)]
ternasPitagoricas n = 
    [(n,a,b) | a <- [1..n-1],
               let b2 = n^2-a^2,
               let b = round (sqrt (fromIntegral b2)),
               b < a,     
               b^2 == b2]

-- 2ª definición
-- =============

pitagoricos2 :: [Integer]
pitagoricos2 = [round (sqrt (fromIntegral a)) | 
                a <- cuadrados,
                let xs = takeWhile (< a) cuadrados,
                any (\b -> a-b `elem` xs) xs]

-- cuadrados es la lista de los cuadrados de los números naturales. Por
-- ejemplo, 
--    take 10 cuadrados  ==  [1,4,9,16,25,36,49,64,81,100]
cuadrados :: [Integer]
cuadrados = [n*n | n <- [1..]]

-- El cálculo de la posición de 2015 es
--    ghci> length (takeWhile (< 2015) pitagoricos)
--    1195

import Test.QuickCheck

-- 1ª definición

-- =============

pitagoricos :: [Integer]

pitagoricos = [n | n <- [1..], esPitagorico n]

-- (esPitagorico n) se verifica si n es pitagórico. Por ejemplo,

-- esPitagorico 5 == True

-- esPitagorico 6 == False

esPitagorico :: Integer -> Bool

esPitagorico n = not (null (ternasPitagoricas n))

-- (ternasPitagoricas n) es la lista de las ternas (n,a,b) tales que

-- n² = a²+b² con b < a < n. Por ejemplo,

-- ghci> ternasPitagoricas 5

-- [(5,4,3)]

-- ghci> ternasPitagoricas 2015

-- [(2015,1612,1209),(2015,1736,1023),(2015,1860,775),(2015,1953,496)]

ternasPitagoricas :: Integer -> [(Integer,Integer,Integer)]

ternasPitagoricas n =

[(n,a,b) | a <- [1..n-1],

let b2 = n^2-a^2,

let b = round (sqrt (fromIntegral b2)),

b < a,

b^2 == b2]

-- 2ª definición

-- =============

pitagoricos2 :: [Integer]

pitagoricos2 = [round (sqrt (fromIntegral a)) |

a <- cuadrados,

let xs = takeWhile (< a) cuadrados,

any (\b -> a-b `elem` xs) xs]

-- cuadrados es la lista de los cuadrados de los números naturales. Por

-- ejemplo,

-- take 10 cuadrados == [1,4,9,16,25,36,49,64,81,100]

cuadrados :: [Integer]

cuadrados = [n*n | n <- [1..]]

-- El cálculo de la posición de 2015 es

-- ghci> length (takeWhile (< 2015) pitagoricos)

-- 1195

2 Comentarios

pitagoricos :: [Integer]
pitagoricos = [n | n <- [1..], not (null (pitagorico n))]

pitagorico :: Integer -> [(Integer,Integer)]
pitagorico n = 
    [(a,b) | b <- [1..n],
             let m = n^2 - b^2, 
             m > 0, 
             let a = round((fromIntegral m)**(1/2)), 
             a < b, 
             a^2 == m]

pitagoricos :: [Integer]

pitagoricos = [n | n <- [1..], not (null (pitagorico n))]

pitagorico :: Integer -> [(Integer,Integer)]

pitagorico n =

[(a,b) | b <- [1..n],

let m = n^2 - b^2,

m > 0,

let a = round((fromIntegral m)**(1/2)),

a < b,

a^2 == m]

Responder

-- La posicion que ocupa 2015 en la lista pitagoricos es 1196:
--  take 1196 pitagoricos
[5,10,13,15,17,20,25,26,29,30,34,35,37,39,40,41,45,50,51,52,53,55,58,60,61,65,68,70,73,74,75,78,80,82,85,87,89,90,91,95,97,100,101,102,104,105,106,109,110,111,113,115,116,117,119,120,122,123,125,130,135,136,137,140,143,145,146,148,149,150,153,155,156,157,159,160,164,165,169,170,173,174,175,178,180,181,182,183,185,187,190,193,194,195,197,200,202,203,204,205,208,210,212,215,218,219,220,221,222,225,226,229,230,232,233,234,235,238,240,241,244,245,246,247,250,255,257,259,260,261,265,267,269,270,272,273,274,275,277,280,281,285,286,287,289,290,291,292,293,295,296,298,299,300,303,305,306,310,312,313,314,315,317,318,319,320,323,325,327,328,330,333,335,337,338,339,340,345,346,348,349,350,351,353,355,356,357,360,362,364,365,366,369,370,371,373,374,375,377,380,385,386,388,389,390,391,394,395,397,400,401,403,404,405,406,407,408,409,410,411,415,416,420,421,424,425,427,429,430,433,435,436,438,440,442,444,445,447,449,450,451,452,455,457,458,459,460,461,464,465,466,468,470,471,475,476,477,480,481,482,485,488,490,492,493,494,495,500,505,507,509,510,511,514,515,518,519,520,521,522,525,527,530,533,534,535,538,540,541,543,544,545,546,548,549,550,551,554,555,557,559,560,561,562,565,569,570,572,574,575,577,578,579,580,582,583,584,585,586,590,591,592,593,595,596,598,600,601,605,606,609,610,611,612,613,615,617,620,623,624,625,626,628,629,630,634,635,636,637,638,640,641,645,646,650,653,654,655,656,657,660,661,663,665,666,667,670,671,673,674,675,676,677,678,679,680,685,687,689,690,692,695,696,697,698,699,700,701,702,703,705,706,707,709,710,712,714,715,720,723,724,725,728,730,731,732,733,735,738,740,741,742,745,746,748,750,754,755,757,760,761,763,765,767,769,770,771,772,773,775,776,777,778,779,780,782,783,785,788,790,791,793,794,795,797,799,800,801,802,803,805,806,807,808,809,810,812,814,815,816,818,819,820,821,822,825,829,830,831,832,833,835,840,841,842,843,845,848,850,851,853,854,855,857,858,860,861,865,866,867,870,871,872,873,875,876,877,879,880,881,884,885,888,890,894,895,897,898,899,900,901,902,904,905,909,910,914,915,916,918,920,922,923,925,928,929,930,932,935,936,937,939,940,941,942,943,945,949,950,951,952,953,954,955,957,959,960,962,964,965,969,970,975,976,977,979,980,981,984,985,986,988,990,995,997,999,1000,1001,1003,1005,1007,1009,1010,1011,1013,1014,1015,1017,1018,1020,1021,1022,1025,1027,1028,1030,1033,1035,1036,1037,1038,1040,1042,1043,1044,1045,1047,1049,1050,1053,1054,1055,1059,1060,1061,1065,1066,1067,1068,1069,1070,1071,1073,1075,1076,1079,1080,1082,1085,1086,1088,1090,1092,1093,1095,1096,1097,1098,1099,1100,1102,1105,1107,1108,1109,1110,1111,1113,1114,1115,1117,1118,1119,1120,1122,1124,1125,1129,1130,1131,1135,1138,1139,1140,1144,1145,1147,1148,1150,1153,1154,1155,1156,1157,1158,1159,1160,1164,1165,1166,1167,1168,1170,1172,1173,1175,1180,1181,1182,1183,1184,1185,1186,1189,1190,1191,1192,1193,1195,1196,1199,1200,1201,1202,1203,1205,1207,1209,1210,1211,1212,1213,1215,1217,1218,1219,1220,1221,1222,1224,1225,1226,1227,1229,1230,1233,1234,1235,1237,1240,1241,1243,1245,1246,1247,1248,1249,1250,1252,1255,1256,1258,1260,1261,1263,1265,1267,1268,1270,1271,1272,1274,1275,1276,1277,1280,1281,1282,1285,1287,1289,1290,1292,1295,1297,1299,1300,1301,1305,1306,1308,1309,1310,1312,1313,1314,1315,1320,1321,1322,1325,1326,1330,1332,1334,1335,1339,1340,1341,1342,1343,1345,1346,1347,1348,1350,1351,1352,1353,1354,1355,1356,1358,1360,1361,1363,1365,1369,1370,1371,1373,1374,1375,1377,1378,1379,1380,1381,1383,1384,1385,1387,1390,1391,1392,1394,1395,1396,1398,1400,1402,1403,1404,1405,1406,1409,1410,1411,1412,1413,1414,1415,1417,1418,1420,1421,1424,1425,1428,1429,1430,1431,1433,1435,1440,1443,1445,1446,1448,1450,1453,1455,1456,1460,1462,1464,1465,1466,1469,1470,1475,1476,1479,1480,1481,1482,1484,1485,1489,1490,1492,1493,1495,1496,1500,1505,1507,1508,1510,1513,1514,1515,1517,1520,1521,1522,1525,1526,1527,1530,1533,1534,1535,1537,1538,1540,1542,1544,1545,1546,1547,1549,1550,1552,1553,1554,1555,1556,1557,1558,1560,1563,1564,1565,1566,1570,1573,1575,1576,1580,1581,1582,1585,1586,1588,1590,1591,1594,1595,1597,1598,1599,1600,1601,1602,1603,1604,1605,1606,1609,1610,1612,1613,1614,1615,1616,1618,1620,1621,1623,1624,1625,1628,1629,1630,1631,1632,1635,1636,1637,1638,1639,1640,1642,1643,1644,1645,1647,1649,1650,1651,1653,1655,1657,1658,1660,1662,1664,1665,1666,1669,1670,1671,1675,1677,1679,1680,1681,1682,1683,1684,1685,1686,1687,1690,1691,1693,1695,1696,1697,1700,1702,1703,1705,1706,1707,1708,1709,1710,1711,1714,1715,1716,1717,1720,1721,1722,1725,1727,1729,1730,1731,1732,1733,1734,1735,1737,1739,1740,1741,1742,1744,1745,1746,1749,1750,1751,1752,1753,1754,1755,1758,1760,1762,1763,1765,1768,1769,1770,1773,1775,1776,1777,1779,1780,1781,1785,1788,1789,1790,1794,1795,1796,1798,1799,1800,1801,1802,1803,1804,1805,1807,1808,1810,1813,1815,1818,1819,1820,1825,1827,1828,1830,1832,1833,1835,1836,1839,1840,1843,1844,1845,1846,1850,1851,1853,1855,1856,1858,1859,1860,1861,1864,1865,1869,1870,1872,1873,1874,1875,1877,1878,1880,1882,1883,1884,1885,1886,1887,1889,1890,1891,1895,1898,1900,1901,1902,1903,1904,1905,1906,1908,1910,1911,1913,1914,1915,1918,1919,1920,1921,1923,1924,1925,1927,1928,1930,1933,1935,1937,1938,1939,1940,1943,1945,1949,1950,1952,1954,1955,1958,1959,1960,1961,1962,1963,1965,1967,1968,1970,1971,1972,1973,1975,1976,1980,1983,1985,1989,1990,1991,1993,1994,1995,1997,1998,2000,2001,2002,2005,2006,2009,2010,2013,2014,2015] --

-- La posicion que ocupa 2015 en la lista pitagoricos es 1196:

-- take 1196 pitagoricos

[5,10,13,15,17,20,25,26,29,30,34,35,37,39,40,41,45,50,51,52,53,55,58,60,61,65,68,70,73,74,75,78,80,82,85,87,89,90,91,95,97,100,101,102,104,105,106,109,110,111,113,115,116,117,119,120,122,123,125,130,135,136,137,140,143,145,146,148,149,150,153,155,156,157,159,160,164,165,169,170,173,174,175,178,180,181,182,183,185,187,190,193,194,195,197,200,202,203,204,205,208,210,212,215,218,219,220,221,222,225,226,229,230,232,233,234,235,238,240,241,244,245,246,247,250,255,257,259,260,261,265,267,269,270,272,273,274,275,277,280,281,285,286,287,289,290,291,292,293,295,296,298,299,300,303,305,306,310,312,313,314,315,317,318,319,320,323,325,327,328,330,333,335,337,338,339,340,345,346,348,349,350,351,353,355,356,357,360,362,364,365,366,369,370,371,373,374,375,377,380,385,386,388,389,390,391,394,395,397,400,401,403,404,405,406,407,408,409,410,411,415,416,420,421,424,425,427,429,430,433,435,436,438,440,442,444,445,447,449,450,451,452,455,457,458,459,460,461,464,465,466,468,470,471,475,476,477,480,481,482,485,488,490,492,493,494,495,500,505,507,509,510,511,514,515,518,519,520,521,522,525,527,530,533,534,535,538,540,541,543,544,545,546,548,549,550,551,554,555,557,559,560,561,562,565,569,570,572,574,575,577,578,579,580,582,583,584,585,586,590,591,592,593,595,596,598,600,601,605,606,609,610,611,612,613,615,617,620,623,624,625,626,628,629,630,634,635,636,637,638,640,641,645,646,650,653,654,655,656,657,660,661,663,665,666,667,670,671,673,674,675,676,677,678,679,680,685,687,689,690,692,695,696,697,698,699,700,701,702,703,705,706,707,709,710,712,714,715,720,723,724,725,728,730,731,732,733,735,738,740,741,742,745,746,748,750,754,755,757,760,761,763,765,767,769,770,771,772,773,775,776,777,778,779,780,782,783,785,788,790,791,793,794,795,797,799,800,801,802,803,805,806,807,808,809,810,812,814,815,816,818,819,820,821,822,825,829,830,831,832,833,835,840,841,842,843,845,848,850,851,853,854,855,857,858,860,861,865,866,867,870,871,872,873,875,876,877,879,880,881,884,885,888,890,894,895,897,898,899,900,901,902,904,905,909,910,914,915,916,918,920,922,923,925,928,929,930,932,935,936,937,939,940,941,942,943,945,949,950,951,952,953,954,955,957,959,960,962,964,965,969,970,975,976,977,979,980,981,984,985,986,988,990,995,997,999,1000,1001,1003,1005,1007,1009,1010,1011,1013,1014,1015,1017,1018,1020,1021,1022,1025,1027,1028,1030,1033,1035,1036,1037,1038,1040,1042,1043,1044,1045,1047,1049,1050,1053,1054,1055,1059,1060,1061,1065,1066,1067,1068,1069,1070,1071,1073,1075,1076,1079,1080,1082,1085,1086,1088,1090,1092,1093,1095,1096,1097,1098,1099,1100,1102,1105,1107,1108,1109,1110,1111,1113,1114,1115,1117,1118,1119,1120,1122,1124,1125,1129,1130,1131,1135,1138,1139,1140,1144,1145,1147,1148,1150,1153,1154,1155,1156,1157,1158,1159,1160,1164,1165,1166,1167,1168,1170,1172,1173,1175,1180,1181,1182,1183,1184,1185,1186,1189,1190,1191,1192,1193,1195,1196,1199,1200,1201,1202,1203,1205,1207,1209,1210,1211,1212,1213,1215,1217,1218,1219,1220,1221,1222,1224,1225,1226,1227,1229,1230,1233,1234,1235,1237,1240,1241,1243,1245,1246,1247,1248,1249,1250,1252,1255,1256,1258,1260,1261,1263,1265,1267,1268,1270,1271,1272,1274,1275,1276,1277,1280,1281,1282,1285,1287,1289,1290,1292,1295,1297,1299,1300,1301,1305,1306,1308,1309,1310,1312,1313,1314,1315,1320,1321,1322,1325,1326,1330,1332,1334,1335,1339,1340,1341,1342,1343,1345,1346,1347,1348,1350,1351,1352,1353,1354,1355,1356,1358,1360,1361,1363,1365,1369,1370,1371,1373,1374,1375,1377,1378,1379,1380,1381,1383,1384,1385,1387,1390,1391,1392,1394,1395,1396,1398,1400,1402,1403,1404,1405,1406,1409,1410,1411,1412,1413,1414,1415,1417,1418,1420,1421,1424,1425,1428,1429,1430,1431,1433,1435,1440,1443,1445,1446,1448,1450,1453,1455,1456,1460,1462,1464,1465,1466,1469,1470,1475,1476,1479,1480,1481,1482,1484,1485,1489,1490,1492,1493,1495,1496,1500,1505,1507,1508,1510,1513,1514,1515,1517,1520,1521,1522,1525,1526,1527,1530,1533,1534,1535,1537,1538,1540,1542,1544,1545,1546,1547,1549,1550,1552,1553,1554,1555,1556,1557,1558,1560,1563,1564,1565,1566,1570,1573,1575,1576,1580,1581,1582,1585,1586,1588,1590,1591,1594,1595,1597,1598,1599,1600,1601,1602,1603,1604,1605,1606,1609,1610,1612,1613,1614,1615,1616,1618,1620,1621,1623,1624,1625,1628,1629,1630,1631,1632,1635,1636,1637,1638,1639,1640,1642,1643,1644,1645,1647,1649,1650,1651,1653,1655,1657,1658,1660,1662,1664,1665,1666,1669,1670,1671,1675,1677,1679,1680,1681,1682,1683,1684,1685,1686,1687,1690,1691,1693,1695,1696,1697,1700,1702,1703,1705,1706,1707,1708,1709,1710,1711,1714,1715,1716,1717,1720,1721,1722,1725,1727,1729,1730,1731,1732,1733,1734,1735,1737,1739,1740,1741,1742,1744,1745,1746,1749,1750,1751,1752,1753,1754,1755,1758,1760,1762,1763,1765,1768,1769,1770,1773,1775,1776,1777,1779,1780,1781,1785,1788,1789,1790,1794,1795,1796,1798,1799,1800,1801,1802,1803,1804,1805,1807,1808,1810,1813,1815,1818,1819,1820,1825,1827,1828,1830,1832,1833,1835,1836,1839,1840,1843,1844,1845,1846,1850,1851,1853,1855,1856,1858,1859,1860,1861,1864,1865,1869,1870,1872,1873,1874,1875,1877,1878,1880,1882,1883,1884,1885,1886,1887,1889,1890,1891,1895,1898,1900,1901,1902,1903,1904,1905,1906,1908,1910,1911,1913,1914,1915,1918,1919,1920,1921,1923,1924,1925,1927,1928,1930,1933,1935,1937,1938,1939,1940,1943,1945,1949,1950,1952,1954,1955,1958,1959,1960,1961,1962,1963,1965,1967,1968,1970,1971,1972,1973,1975,1976,1980,1983,1985,1989,1990,1991,1993,1994,1995,1997,1998,2000,2001,2002,2005,2006,2009,2010,2013,2014,2015] --

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2015 y los números pitagóricos

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