求 1^2+2^2+3^2+...+n^2 = m^2 的正整数解

medie2005

其实有个很快的办法搜其他的解（如果存在的话）。
n=6p^2
n+1=q^2
2n+1=r^2
选定两个式子，联立得到一个二次不定方程，解这个不定方程，可以得到一个递推式。然后，只要判断另外一个式子满足不满足就可以了。
不过，我倾向于只有一组解。

无心人

老大上来这么晚
周末有活动？
还是给国家做贡献了
去义务修路了？

medie2005

呵呵,我都听不懂你在说什么了.
明天是由活动,陪老妈出去玩.

无心人

呵呵

无心人

我这里有本书推荐是测试
65, 64, 63, 11四个模
概率是$6/715$

无心人

这里是平方剩余数量， 和对应的模（第二个是模）
[(2,3),(2,4),(3,5),(3,8),(4,6),(4,7),(4,9),(4,12),(4,16),(6,10),(6,11),(6,15),(6
,20),(6,24),(7,13),(7,32),(8,14),(8,18),(8,21),(8,28),(8,36),(8,48),(9,17),(9,40
),(10,19),(11,25),(11,27),(12,22),(12,23),(12,30),(12,33),(12,35),(12,44),(12,45
),(12,56),(12,60),(12,64),(12,72),(12,80),(14,26),(14,39),(14,52),(14,96),(15,29
),(16,31),(16,42),(16,63),(16,84),(18,34),(18,51),(18,55),(18,68),(18,88),(19,37
),(20,38),(20,57),(20,76),(21,41),(21,65),(22,43),(22,49),(22,50),(22,54),(22,75
),(22,100),(24,46),(24,47),(24,66),(24,69),(24,70),(24,77),(24,90),(24,92),(24,9
9),(27,53),(27,85),(28,78),(28,91),(30,58),(30,59),(30,87),(30,95),(31,61),(31,8
1),(32,62),(32,93),(34,67),(36,71),(37,73),(38,74),(40,79),(42,82),(42,83),(44,8
6),(44,98),(45,89),(48,94),(49,97)]

无心人

下面是1000内的结果
[(2,3),(2,4),(3,5),(3,8),(4,6),(4,7),(4,9),(4,12),(4,16),(6,10),(6,11),(6,15),(6
,20),(6,24),(7,13),(7,32),(8,14),(8,18),(8,21),(8,28),(8,36),(8,48),(9,17),(9,40
),(10,19),(11,25),(11,27),(12,22),(12,23),(12,30),(12,33),(12,35),(12,44),(12,45
),(12,56),(12,60),(12,64),(12,72),(12,80),(14,26),(14,39),(14,52),(14,96),(15,29
),(16,31),(16,42),(16,63),(16,84),(16,112),(16,144),(18,34),(18,51),(18,55),(18,
68),(18,88),(18,120),(19,37),(20,38),(20,57),(20,76),(21,41),(21,65),(21,104),(2
1,160),(22,43),(22,49),(22,50),(22,54),(22,75),(22,100),(22,108),(23,128),(24,46
),(24,47),(24,66),(24,69),(24,70),(24,77),(24,90),(24,92),(24,99),(24,105),(24,1
32),(24,140),(24,168),(24,176),(24,180),(24,192),(24,240),(27,53),(27,85),(27,13
6),(28,78),(28,91),(28,117),(28,156),(28,208),(28,224),(28,288),(30,58),(30,59),
(30,87),(30,95),(30,116),(30,152),(31,61),(31,81),(32,62),(32,93),(32,124),(32,1
26),(32,252),(32,336),(33,135),(33,200),(33,216),(34,67),(36,71),(36,102),(36,11
0),(36,115),(36,119),(36,153),(36,165),(36,184),(36,204),(36,220),(36,264),(36,2
72),(36,280),(36,320),(36,360),(37,73),(38,74),(38,111),(38,148),(40,79),(40,114
),(40,133),(40,171),(40,228),(40,304),(42,82),(42,83),(42,123),(42,130),(42,143)
,(42,164),(42,195),(42,260),(42,312),(42,352),(42,480),(44,86),(44,98),(44,129),
(44,147),(44,150),(44,172),(44,175),(44,189),(44,196),(44,225),(44,256),(44,300)
,(44,400),(44,432),(45,89),(45,145),(45,232),(46,384),(48,94),(48,138),(48,141),
(48,154),(48,155),(48,161),(48,188),(48,198),(48,207),(48,210),(48,231),(48,248)
,(48,276),(48,308),(48,315),(48,368),(48,396),(48,420),(48,448),(48,504),(48,528
),(48,560),(48,576),(48,720),(49,97),(49,416),(51,101),(52,103),(53,125),(54,106
),(54,107),(54,159),(54,170),(54,187),(54,212),(54,255),(54,340),(54,408),(54,44
0),(55,109),(56,121),(56,182),(56,234),(56,273),(56,364),(56,468),(56,624),(56,6
72),(57,113),(57,185),(57,296),(60,118),(60,174),(60,177),(60,190),(60,203),(60,
209),(60,236),(60,261),(60,285),(60,348),(60,380),(60,456),(60,464),(62,122),(62
,162),(62,183),(62,244),(62,324),(63,205),(63,221),(63,328),(63,520),(63,544),(6
4,127),(64,186),(64,217),(64,279),(64,372),(64,496),(66,131),(66,215),(66,245),(
66,270),(66,275),(66,297),(66,344),(66,392),(66,540),(66,600),(68,134),(68,201),
(68,268),(69,137),(69,640),(70,139),(70,247),(70,608),(72,142),(72,213),(72,230)
,(72,235),(72,238),(72,253),(72,284),(72,306),(72,330),(72,345),(72,357),(72,376
),(72,385),(72,460),(72,476),(72,495),(72,552),(72,612),(72,616),(72,660),(72,70
4),(72,792),(72,816),(72,840),(72,880),(72,960),(74,146),(74,219),(74,292),(75,1
49),(76,151),(76,222),(76,259),(76,333),(76,444),(76,592),(77,325),(77,351),(77,
800),(77,864),(79,157),(79,169),(80,158),(80,237),(80,266),(80,316),(80,342),(80
,399),(80,532),(80,684),(80,912),(81,265),(81,424),(81,680),(82,163),(84,166),(8
4,167),(84,246),(84,249),(84,286),(84,287),(84,299),(84,332),(84,369),(84,390),(
84,429),(84,455),(84,492),(84,572),(84,585),(84,656),(84,728),(84,736),(84,780),
(84,832),(84,936),(87,173),(87,512),(88,258),(88,294),(88,301),(88,350),(88,378)
,(88,387),(88,441),(88,450),(88,516),(88,525),(88,588),(88,688),(88,700),(88,756
),(88,768),(88,784),(88,900),(90,178),(90,179),(90,267),(90,290),(90,295),(90,31
9),(90,323),(90,356),(90,435),(90,472),(90,580),(90,696),(90,760),(91,181),(92,2
43),(92,896),(93,305),(93,405),(93,488),(93,648),(96,191),(96,282),(96,310),(96,
322),(96,329),(96,341),(96,414),(96,423),(96,462),(96,465),(96,483),(96,564),(96
,620),(96,630),(96,644),(96,693),(96,744),(96,752),(96,828),(96,924),(97,193),(9
8,194),(98,291),(98,388),(99,197),(99,425),(99,459),(100,199),(102,202),(102,303
),(102,335),(102,404),(102,536),(104,206),(104,309),(104,412),(105,377),(105,928
),(106,211),(106,250),(106,375),(106,500),(108,214),(108,318),(108,321),(108,355
),(108,371),(108,374),(108,391),(108,428),(108,477),(108,510),(108,561),(108,568
),(108,595),(108,636),(108,748),(108,765),(108,848),(108,920),(108,952),(110,218
),(110,327),(110,436),(110,475),(110,513),(111,365),(111,584),(112,223),(112,242
),(112,363),(112,403),(112,484),(112,546),(112,819),(112,992),(114,226),(114,227
),(114,339),(114,370),(114,407),(114,452),(114,555),(114,740),(114,888),(115,229
),(117,233),(120,239),(120,354),(120,395),(120,406),(120,413),(120,418),(120,437
),(120,522),(120,531),(120,570),(120,609),(120,627),(120,632),(120,665),(120,708
),(120,812),(120,836),(120,855),(120,944),(121,241),(121,675),(124,366),(124,427
),(124,549),(124,567),(124,732),(124,976),(126,251),(126,410),(126,415),(126,442
),(126,451),(126,615),(126,663),(126,664),(126,715),(126,820),(126,884),(126,984
),(128,254),(128,381),(128,434),(128,508),(128,558),(128,651),(128,868),(129,257
),(132,262),(132,263),(132,393),(132,430),(132,473),(132,490),(132,524),(132,539
),(132,550),(132,575),(132,594),(132,621),(132,645),(132,735),(132,825),(132,860
),(132,945),(132,980),(133,481),(135,269),(135,445),(135,493),(135,712),(136,271
),(136,402),(136,469),(136,603),(136,804),(137,289),(138,274),(138,411),(138,548
),(139,277),(140,278),(140,417),(140,494),(140,556),(140,741),(140,988),(141,281
),(142,283),(144,426),(144,470),(144,497),(144,506),(144,517),(144,527),(144,639
),(144,690),(144,705),(144,714),(144,759),(144,770),(144,805),(144,852),(144,940
),(144,990),(147,293),(147,485),(147,533),(147,776),(148,438),(148,511),(148,657
),(148,876),(150,298),(150,447),(150,551),(150,596),(151,343),(152,302),(152,453
),(152,518),(152,604),(152,666),(152,777),(153,505),(153,808),(154,307),(154,559
),(154,637),(154,650),(154,702),(154,975),(156,311),(156,515),(156,824),(157,313
),(158,314),(158,338),(158,471),(158,507),(158,628),(158,676),(159,317),(159,100
0),(160,474),(160,553),(160,589),(160,711),(160,798),(160,948),(162,530),(162,53
5),(162,583),(162,795),(162,856),(162,935),(164,326),(164,489),(164,652),(165,54
5),(165,725),(165,783),(165,872),(166,331),(168,334),(168,498),(168,501),(168,57
4),(168,581),(168,598),(168,605),(168,611),(168,668),(168,738),(168,747),(168,85
8),(168,861),(168,897),(168,910),(168,968),(168,996),(169,337),(171,565),(171,62
9),(171,904),(172,361),(174,346),(174,347),(174,519),(174,692),(175,349),(176,60
2),(176,774),(176,775),(176,837),(176,882),(176,903),(177,353),(180,358),(180,35
9),(180,534),(180,537),(180,590),(180,623),(180,638),(180,646),(180,649),(180,66
7),(180,716),(180,801),(180,870),(180,885),(180,957),(180,969),(182,362),(182,54
3),(182,724),(184,367),(184,486),(184,972),(186,610),(186,671),(186,810),(186,89
1),(186,915),(187,373),(189,689),(189,697),(190,379),(190,703),(192,382),(192,38
3),(192,573),(192,635),(192,658),(192,682),(192,713),(192,764),(192,846),(192,93
0),(192,966),(192,987),(194,386),(194,579),(194,772),(195,389),(196,582),(196,67
9),(196,873),(198,394),(198,591),(198,655),(198,731),(198,788),(198,833),(198,85
0),(198,918),(199,397),(200,398),(200,597),(200,796),(201,401),(204,606),(204,67
0),(204,707),(204,737),(204,909),(205,409),(207,685),(208,618),(208,721),(208,92
7),(209,925),(209,999),(210,419),(210,695),(210,754),(210,767),(210,779),(211,42
1),(212,422),(212,633),(212,750),(212,844),(212,875),(216,431),(216,642),(216,71
0),(216,742),(216,749),(216,781),(216,782),(216,799),(216,954),(216,963),(217,43
3),(217,793),(220,439),(220,654),(220,763),(220,817),(220,931),(220,950),(220,98
1),(222,443),(222,730),(222,803),(224,446),(224,669),(224,726),(224,806),(224,84
7),(224,892),(225,449),(225,745),(228,454),(228,678),(228,681),(228,755),(228,79
1),(228,814),(228,851),(228,908),(229,457),(230,458),(230,687),(230,916),(231,46
1),(232,463),(234,466),(234,467),(234,699),(234,932),(237,785),(237,845),(238,87
1),(240,478),(240,479),(240,717),(240,790),(240,826),(240,869),(240,874),(240,89
3),(240,899),(240,956),(242,482),(242,723),(242,964),(243,901),(244,487),(246,49
1),(246,815),(248,854),(250,499),(252,502),(252,503),(252,753),(252,830),(252,83
5),(252,902),(252,913),(252,923),(252,943),(254,529),(255,509),(256,762),(256,88
9),(258,514),(258,771),(259,949),(261,521),(261,625),(261,865),(262,523),(264,52
6),(264,786),(264,789),(264,917),(264,946),(264,989),(266,962),(270,538),(270,80
7),(270,890),(270,895),(270,979),(270,986),(271,541),(272,542),(272,813),(272,93
8),(273,905),(274,547),(274,578),(274,729),(274,867),(276,822),(276,959),(278,55
4),(278,831),(279,557),(280,834),(280,973),(282,562),(282,563),(282,843),(284,56
6),(284,849),(285,569),(286,571),(288,955),(288,994),(289,577),(291,965),(294,58
6),(294,587),(294,879),(294,970),(297,593),(297,985),(300,599),(300,894),(300,99
5),(301,601),(302,686),(304,607),(304,906),(307,613),(308,614),(308,921),(309,61
7),(310,619),(312,622),(312,933),(314,626),(314,939),(316,631),(316,942),(318,63
4),(318,951),(321,641),(322,643),(324,647),(327,653),(328,978),(330,659),(331,66
1),(332,662),(332,993),(337,673),(338,674),(339,677),(342,683),(344,722),(346,69
1),(348,694),(350,698),(351,701),(354,706),(355,709),(360,718),(360,719),(364,72
7),(367,733),(368,734),(370,739),(372,743),(374,746),(376,751),(379,757),(380,75
8),(381,761),(384,766),(385,769),(387,773),(390,778),(394,787),(398,794),(399,79
7),(402,802),(405,809),(406,811),(407,841),(410,818),(411,821),(412,823),(414,82
7),(415,829),(420,838),(420,839),(422,842),(427,853),(429,857),(430,859),(432,86
2),(432,863),(434,866),(439,877),(440,878),(441,881),(442,883),(444,886),(444,88
7),(450,898),(454,907),(456,911),(458,914),(460,919),(462,922),(464,926),(465,92
9),(466,961),(468,934),(469,937),(471,941),(474,947),(477,953),(480,958),(484,96
7),(486,971),(488,974),(489,977),(492,982),(492,983),(496,991),(499,997),(500,99
8)]

无心人

可以看到
11， 63， 64， 65几乎是最好的测试参数了

wayne

可以证明只有一种情况：
n=6p^2  =6（2k）^2
n+1=q^2 = (2x+1)^2
2n+1=r^2 =(2y+1)^2
将 n=24k^2代入下面的两个式子得
6k^2 =(y-x)(y+x+1)

因为(y-x)，(y+x+1)必是一奇一偶，所以

winxos

原帖由 无心人 于 2009-2-22 13:18 发表
可以看到
11， 63， 64， 65几乎是最好的测试参数了

还是第一次听说过平方剩余
用处仅仅是用来快速测试平方数？
而且这种测试概率型的，通过了还是要仔细测的。