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。