任意自然数

northwolves

证明任意自然数n都可表示为：

　　a²+b
n = ———　的形式，a，b均为自然数
　　a+b²

KeyTo9_Fans

n=1: a=1, b=1.
n=2: a=5, b=3.
n=3: a=5, b=2.
n=4: a=10, b=4.
n=5: a=27, b=11.
n=6: a=69, b=27.
n=7: a=12, b=3.
n=8: a=38, b=12.
n=9: a=20, b=5.
n=10: a=103, b=31.
n=11: a=14, b=2.
n=12: a=335, b=95.
n=13: a=19, b=3.
n=14: a=1859, b=495.
n=15: a=147, b=36.
n=16: a=37, b=7.
n=17: a=293, b=69.
n=18: a=44, b=8.
n=19: a=54, b=10.
n=20: a=1043, b=231.
n=21: a=89, b=17.
n=22: a=50, b=8.
n=23: a=38, b=5.
n=24: a=404, b=80.
n=25: a=36, b=4.
n=26: a=33, b=3.
n=27: a=367, b=68.
n=28: a=263, b=47.
n=29: a=45, b=5.
n=30: a=77, b=11.
n=31: a=84, b=12.
n=32: a=147, b=23.
n=33: a=12350, b=2147.
n=34: a=129, b=19.
n=35: a=57, b=6.
n=36: a=1962, b=324.
n=37: a=49, b=4.
n=38: a=37665, b=6107.
n=39: a=109, b=14.
n=40: a=217, b=31.
n=41: a=42185, b=6585.
n=42: a=5225, b=803.
n=43: a=103, b=12.
n=44: a=1110, b=164.
n=45: a=119, b=14.
n=46: a=101, b=11.
n=47: a=65, b=5.
n=48: a=1770, b=252.
n=49: a=550, b=75.
n=50: a=2507, b=351.
n=51: a=88, b=8.
n=52: a=1180, b=160.
n=53: a=153, b=17.
n=54: a=9683509, b=1317755.
n=55: a=171, b=19.
n=56: a=122, b=12.
n=57: a=407, b=50.
n=58: a=214, b=24.
n=59: a=15699, b=2040.
n=60: a=44577, b=5751.
n=61: a=80, b=5.
n=62: a=191015, b=24255.
n=63: a=170, b=17.
n=64: a=8224, b=1024.
n=65: a=4233, b=521.
n=66: a=218139, b=26847.
n=67: a=75, b=3.
n=68: a=300, b=32.
n=69: a=5265595, b=633899.
n=70: a=2244, b=264.
n=71: a=3246, b=381.
n=72: a=16455, b=1935.
n=73: a=270, b=27.
n=74: a=855, b=95.
n=75: a=40273, b=4646.
n=76: a=118600, b=13600.
n=77: a=1250, b=138.
n=78: a=174793, b=19787.
n=79: a=170, b=14.
n=80: a=5192, b=576.
n=81: a=101, b=5.
n=82: a=6733, b=739.
n=83: a=264, b=24.
n=84: a=230, b=20.
n=85: a=12913, b=1396.
n=86: a=297, b=27.
n=87: a=138, b=9.
n=88: a=123, b=7.
n=89: a=22276589, b=2361309.
n=90: a=26333, b=2771.
n=91: a=307, b=27.
n=92: a=197, b=15.
n=93: a=1089, b=108.
n=94: a=824, b=80.
n=95: a=780, b=75.
n=96: a=34527, b=3519.
n=97: a=2649, b=264.
n=98: a=1505, b=147.
n=99: a=8109, b=810.
n=100: a=114, b=4.
n=101: a=10211, b=1011.
n=102: a=649, b=59.
n=103: a=4294, b=418.
n=104: a=222410, b=21804.
n=105: a=145262, b=14171.
n=106: a=226, b=16.
n=107: a=152, b=8.
n=108: a=299, b=23.
n=109: a=3020, b=284.
n=110: a=40109, b=3819.
n=111: a=1121, b=101.
n=112: a=34927, b=3295.
n=113: a=665, b=57.
n=114: a=1969, b=179.
n=115: a=1935, b=175.
n=116: a=450, b=36.
n=117: a=344385, b=31833.
n=118: a=164, b=8.
n=119: a=6005, b=545.
n=120: a=12110, b=1100.
n=121: a=150, b=6.
n=122: a=14895, b=1343.
n=123: a=7570, b=677.
n=124: a=5129, b=455.
n=125: a=231, b=14.
n=126: a=257373, b=22923.
n=127: a=2690, b=233.
n=128: a=2780, b=240.
n=129: a=335404, b=29525.
n=130: a=1879, b=159.
n=131: a=695484, b=60759.
n=132: a=850, b=68.
n=133: a=220, b=12.
n=134: a=58224, b=5024.
n=135: a=212, b=11.
n=136: a=5841, b=495.
n=137: a=290, b=18.
n=138: a=257183, b=21887.
n=139: a=304, b=19.
n=140: a=1208, b=96.
n=141: a=1075494, b=90567.
n=142: a=1122, b=88.
n=143: a=17435, b=1452.
n=144: a=62280, b=5184.
n=145: a=21037, b=1741.
n=146: a=372, b=24.
n=147: a=1173020, b=96743.
n=148: a=22264, b=1824.
n=149: a=920, b=69.
n=150: a=305955, b=24975.
n=151: a=181, b=6.
n=152: a=176970, b=14348.
n=153: a=26349, b=2124.
n=154: a=325, b=19.
n=155: a=177003, b=14211.
n=156: a=6261, b=495.
n=157: a=685, b=48.
n=158: a=675, b=47.
n=159: a=765, b=54.
n=160: a=2295, b=175.
n=161: a=19299851, b=1521035.
n=162: a=562583502, b=44200728.
n=163: a=185, b=5.
n=164: a=589075205, b=45999039.
n=165: a=464, b=29.
n=166: a=8020, b=616.
n=167: a=12102, b=930.
n=168: a=24348, b=1872.
n=169: a=6715, b=510.
n=170: a=28913, b=2211.
n=171: a=14627, b=1112.
n=172: a=362, b=20.
n=173: a=216332055, b=16447415.
n=174: a=314846685, b=23868459.
n=175: a=5922, b=441.
n=176: a=4427, b=327.
n=177: a=407, b=23.
n=178: a=630, b=40.
n=179: a=49445, b=3689.
n=180: a=4020169, b=299639.
n=181: a=1694, b=119.
n=182: a=114425, b=8475.
n=183: a=468, b=27.
n=184: a=19354, b=1420.
n=185: a=458, b=26.
n=186: a=44649, b=3267.
n=187: a=157422, b=11505.
n=188: a=28929, b=2103.
n=189: a=494394, b=35955.
n=190: a=1972, b=136.
n=191: a=275, b=11.
n=192: a=1563930, b=112860.
n=193: a=784, b=49.
n=194: a=6390929, b=458835.
n=195: a=514, b=29.
n=196: a=134554, b=9604.
n=197: a=572, b=33.
n=198: a=560, b=32.
n=199: a=13205, b=929.
n=200: a=1701908, b=120336.
n=201: a=286, b=11.

n=202表示不出来。

geslon

是否有可能 Key版主 编程搜索的值还不够大？

KeyTo9_Fans

正如楼上所说，扩大搜索范围后，n=202表示出来了。

n=202: a=1993602325, b=140269363.
n=203: a=573248, b=40227.
n=204: a=25240, b=1760.
n=205: a=558, b=31.
n=206: a=493111787, b=34356719.
n=207: a=27555, b=1908.
n=208: a=804, b=48.
n=209: a=7273078539, b=503089355.
n=210: a=153873, b=10611.
n=211: a=252, b=7.
n=212: a=29445, b=2015.
n=213: a=247469, b=16949.
n=214: a=474, b=24.
n=215: a=363219, b=24764.
n=216: a=729179, b=49607.
n=217: a=1424, b=89.
n=218: a=6134, b=408.
n=219: a=1047765, b=70794.
n=220: a=625429, b=42159.
n=221: a=284, b=9.
n=222: a=226273, b=15179.
n=223: a=21959, b=1463.
n=224: a=44114, b=2940.
n=225: a=11978, b=791.
n=226: a=278, b=8.
n=227: a=353, b=14.
n=228: a=4554159, b=301599.
n=229: a=244, b=4.
n=230: a=972, b=56.
n=231: a=1169867, b=76964.
n=232: a=485, b=23.
n=233: a=1070, b=62.
n=234: a=665, b=35.
n=235: a=323, b=11.
n=236: a=2862516174, b=186333924.
n=237: a=1172, b=68.
n=238: a=6908, b=440.
n=239: a=138545, b=8954.
n=240: a=202739, b=13079.
n=241: a=1291, b=75.
n=242: a=15180, b=968.
n=243: a=413, b=17.
n=244: a=372749, b=23855.
n=245: a=119097, b=7601.
n=246: a=108957, b=6939.
n=247: a=1512, b=88.
n=248: a=350, b=12.
n=249: a=9104, b=569.
n=250: a=1648, b=96.
n=251: a=5862, b=362.
n=252: a=3583685, b=225743.
n=253: a=295, b=7.
n=254: a=530, b=24.
n=255: a=57615, b=3600.
n=256: a=4162, b=252.
n=257: a=1083, b=59.
n=258: a=15046835, b=936767.
n=259: a=25300, b=1564.
n=260: a=3857, b=231.
n=261: a=2396, b=140.
n=262: a=5555, b=335.
n=263: a=408, b=15.
n=264: a=986309, b=60695.
n=265: a=2838, b=166.
n=266: a=110417889, b=6770147.
n=267: a=290, b=5.
n=268: a=3280, b=192.
n=269: a=489, b=20.
n=270: a=6692850449, b=407313899.
n=271: a=4671250, b=283750.
n=272: a=262415, b=15903.
n=273: a=603, b=27.
n=274: a=42910, b=2584.
n=275: a=1818, b=101.
n=276: a=2385, b=135.
n=277: a=355, b=10.
n=278: a=1542441729, b=92509499.
n=279: a=571901519, b=34238840.
n=280: a=123949, b=7399.
n=281: a=414, b=14.
n=282: a=10352, b=608.
n=283: a=388, b=12.
n=284: a=1040940, b=61760.
n=285: a=18308, b=1076.
n=286: a=24496, b=1440.
n=287: a=36872, b=2168.
n=288: a=74000, b=4352.
n=289: a=19865, b=1160.
n=290: a=84117, b=4931.
n=291: a=357, b=9.
n=292: a=4318, b=244.
n=293: a=1805335115, b=105468795.
n=294: a=302610, b=17640.
n=295: a=2488, b=136.
n=296: a=402, b=12.
n=297: a=4115, b=230.
n=298: a=1538, b=80.
n=299: a=434, b=14.
n=300: a=11773, b=671.
n=301: a=626, b=26.
n=302: a=7017, b=395.
n=303: a=6559, b=368.
n=304: a=41370, b=2364.
n=305: a=33527, b=1911.
n=306: a=95874, b=5472.
n=307: a=2052, b=108.
n=308: a=33860969, b=1929399.
n=309: a=4395105, b=250020.
n=310: a=4577, b=251.
n=311: a=10244, b=572.
n=312: a=2070, b=108.
n=313: a=609, b=24.
n=314: a=61633575, b=3478175.
n=315: a=902, b=41.
n=316: a=3699, b=199.
n=317: a=14670, b=815.
n=318: a=3254725, b=182507.
n=319: a=2469, b=129.
n=320: a=1439186, b=80444.
n=321: a=838741, b=46805.
n=322: a=14948, b=824.
n=323: a=2145, b=110.
n=324: a=472554, b=26244.
n=325: a=105643, b=5851.
n=326: a=677, b=27.
n=327: a=17072, b=935.
n=328: a=2759, b=143.
n=329: a=594105, b=32745.
n=330: a=3512197, b=193331.
n=331: a=2609, b=134.

n=332貌似在b<2^31的范围内无解，只好再次扩大搜索范围，继续在2^31到2^32之间搜寻，终于在b=2952389663时找到了符合条件的a值。

n=332: a=53795100015, b=2952389663

所以只好相信楼主并没有撒谎，命题确实是成立的，接下来要考虑如何证明了……

geslon

哈哈，“撒谎”。

无心人

=> n(a + b^2) = a^2 + b => a^2 - na +b - nb^2 = 0
=> (n^2 - 4(b - nb^2)) = k^2

geslon

命题描述简单，证明还真的费脑筋呢。
在整数范围内，当b=0和b=-1的时候，解是显然的。在自然数范围，难搞。

medie2005

参考一下pell方程理论，应可找到解决途径。

wayne

恒等式 ：$n^3-(2a-n)^2 n+(2n b-1)^2-1=-4 n (a^2+b-(a +b^2) n)$

根据独立性，不妨设A=2a-n,B=2n b-1，那么，问题等价于方程$n^3-A^2 n+B^2-1=0$对于任意的自然数n都有A,B的自然数解

mathe

上面方程可以变化为:
$n*(2a-n)^2-(2n*b-1)^2=n^3-1$
由于我们知道方程
$n*A^2-B^2=n^3-1$有一组解A=n,B=1
而Pell方程$n*X^2-Y^2=-1$总是有无穷组解$(X_t,Y_t)$,其中两个数都是正整数，可以任意大
于是我们展开$(n*sqrt(n)+1)(X_t*sqrt(n)+Y_t)=(n*Y_t+X_t)sqrt(n)+(n^2*X_t+Y_t)$
得到一组解$A=n*Y_t+X_t,B=n^2*X_t+Y_t$
此外我们还需要额外条件$2|A+n,2n|B+1$,
既$2|n*(Y_t+1)+X_t,2n|n^2*X_t+Y_t+1$才行。