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1、与数列有关的不定方程的整数解问题初探 林伟民 （江苏省丹阳市第五中学 212300）数列是高中数学的重要内容，又是学习高等数学的基础，在高考中占有极其重要的地位。2008年高考江苏卷第19题则以数列为载体，综合运用数列与不定方程知识解决问题，使数列与不定方程的整数解问题成为一个新的热点。这类问题对数学思维能力和探索能力提出了更高的要求，因此在近年来的各省市高考模拟卷中，这类问题屡见不鲜。对于一般的不定方程，通常是没有统一的解法，况且有些不定方程我们还无法判别它是否存在整数解，故在这里对于一般不定方程的整数解的求法不作具体讨论。本文着重对与数列有关的不定方程的整数解问题的解法作初步的探讨。1.分
2、类逐一探讨，道尽“不定”悬念在不具备直接求未知数的条件时，利用分类讨论的方法对可能的情况进行逐一讨论，最终求得未知数的值， 是解决数列中不定方程问题的常用策略。例1已知正整数不超过2000，且能表示成不少于60个连续正整数之和，那么这样的的个数是 。【分析】设连续正整数的首项为，项数为，首项为的连续个正整数之和为，则：，等式中均未知，可看作不定方程，由题设条件显然有、，且，考虑先运用个正整数的和不超过2000这个条件，缩小或的探求范围。【解】，得,解得，故最多可取三个值，下面依次进行讨论：（1）当时，解得，或2或3时，或1890或1950；（2）当时，可得，或2时，或；（3）当时，可得,.所以
3、可取6个值：1830，1890，1950，1891，1952，1953.【点评】本题先利用不等关系缩小了未知数的范围，然后对所有可能的情形逐一进行了探讨，揭开了“不定”的神秘面纱，道尽了“不定”的悬念。2.熟用数论常识，化解“不定”难点2.1利用数或式的分解先将不定方程两边的数分解为质因数的乘积（多项式分解为若干个因式的乘积），再利用奇偶性进行讨论。例2由连续正整数组成的数列之和为1000，试求出所有这样的数列。【分析】设数列的首项为，共有项，由数列项之和为1000得如下关于和的不定方程：，可考虑先将1000分解为质因数的乘积，再对左边的数进行奇偶性讨论。【解】将不定方程化为：因为是个奇数，故
4、与的奇偶性相反，由上式知，只属于与中的一个.又因为，从而的值只能为1,5,.这里不能取,否则=相矛盾.将与的可能的取值列表如下：1510001982855由上表可知，所求数列共由3个：时,数列为198，199，200，201，202；时,数列为28，29，30，201，52； 时,数列为55，56，57，70.【点评】此题为比较典型的二元不定方程，其解法也是解不定方程的典型解法，先将右边的数分解为质因数的乘积，再利用左边两个数奇偶性相反且具有确定的大小关系的条件进行分类讨论求解。2.2利用整除的性质由于构成不定方程的数列中的元素均为整数，可利用整除的性质求整数解。例3已知等差数列的首项为，公差
5、为，等比数列的首项为，公比为(其中均为正整数)若，且至少存在三个不同的值使得等式成立，对最小的t试求、的值 【分析】由于数列和中的所有项均为整数，在利用条件得出不定方程后，可用数的整除性求解；而不等式条件则可用来夹逼出整数解。【解】由得：，由得：；由得：，而，即：，从而得：，当时，不合题意，故舍去，所以满足条件的 又，故，即： 若，则，不合题意； 若，则，由于可取到一切整数值，且，故要至少存在三个使得成立，必须整数至少有三个大于或等于3的不等的因数，故满足条件的最小整数为12，所以的最小值为，此时或或12. 【点评】由于均为整数，对等式的整除性讨论是本题的关键，而恰是上述整除性分析成功化解了“
6、不定”的难点。3.妙用不等关系，缩小“不定”空间不定方程的整数解较难确定时，可利用不等式前后夹逼得到整数解。例4各项为实数的等差数列的公差为4，其首项的平方与其余各项之和不超过100，这样的数列至多有 项。【分析】此等差数列的首项和项数均未知，要确定的最大值，只能利用首项的平方与其余各项之和不超过100这个不等式寻机缩小的范围。【解】设是公差为4的等差数列，则（*）当且仅当时，至少存在一个实数满足上面的不等式。因为故.例如：时，故能取到8.4.活用函数工具，实现“不定”转变 关于数列的不定方程的两边均可以看做一个以某变量为主元的函数，通过函数工具，分别研究这两个函数的性质，从而实现“方程”到“
7、函数”的转变。例5数列中,（）求数列的通项公式；（）当为某等差数列的第1项，第项，第+7项，且，求与；【解析】（）过程略，结论为: （）当时，则该等差数列的公差为，即 又，所以，即 和两个等式均为不定方程，其中的和均为整数，可对的取值进行估计。由知，为整数或分母为7的既约分数；由知，为整数或分母为2的既约分数，由于要同时满足和，从而必为整数。由知，结合得，所以只能取7，故， 又由得，下面讨论关于的方程有没有整数解。设则因为，于是当时，从而，在上单调递增.则由，知在上无解.又，或.综上所述,当,且或时满足条件。数列与不定方程（函数或不等式）的交汇使得高考压轴题变化多样，精彩纷呈，解法也有很大的灵
8、活性，以上仅列举了几种常用的探求方法，具体问题还需具体分析，根据题设条件灵活处理。本文发表在苏州大学中学数学月刊2010年第10期Editors note: Judson Jones is a meteorologist, journalist and photographer. He has freelanced with CNN for four years, covering severe weather from tornadoes to typhoons. Follow him on Twitter: jnjonesjr (CNN) - I will always wonder wh
9、at it was like to huddle around a shortwave radio and through the crackling static from space hear the faint beeps of the worlds first satellite - Sputnik. I also missed watching Neil Armstrong step foot on the moon and the first space shuttle take off for the stars. Those events were way before my
10、time.As a kid, I was fascinated with what goes on in the sky, and when NASA pulled the plug on the shuttle program I was heartbroken. Yet the privatized space race has renewed my childhood dreams to reach for the stars.As a meteorologist, Ive still seen many important weather and space events, but r
11、ight now, if you were sitting next to me, youd hear my foot tapping rapidly under my desk. Im anxious for the next one: a space capsule hanging from a crane in the New Mexico desert.Its like the set for a George Lucas movie floating to the edge of space.You and I will have the chance to watch a man
12、take a leap into an unimaginable free fall from the edge of space - live.The (lack of) air up there Watch man jump from 96,000 feet Tuesday, I sat at work glued to the live stream of the Red Bull Stratos Mission. I watched the balloons positioned at different altitudes in the sky to test the winds,
13、knowing that if they would just line up in a vertical straight line we would be go for launch.I feel this mission was created for me because I am also a journalist and a photographer, but above all I live for taking a leap of faith - the feeling of pushing the envelope into uncharted territory.The g
14、uy who is going to do this, Felix Baumgartner, must have that same feeling, at a level I will never reach. However, it did not stop me from feeling his pain when a gust of swirling wind kicked up and twisted the partially filled balloon that would take him to the upper end of our atmosphere. As soon
15、 as the 40-acre balloon, with skin no thicker than a dry cleaning bag, scraped the ground I knew it was over.How claustrophobia almost grounded supersonic skydiverWith each twist, you could see the wrinkles of disappointment on the face of the current record holder and capcom (capsule communications
16、), Col. Joe Kittinger. He hung his head low in mission control as he told Baumgartner the disappointing news: Mission aborted.The supersonic descent could happen as early as Sunday.The weather plays an important role in this mission. Starting at the ground, conditions have to be very calm - winds le
17、ss than 2 mph, with no precipitation or humidity and limited cloud cover. The balloon, with capsule attached, will move through the lower level of the atmosphere (the troposphere) where our day-to-day weather lives. It will climb higher than the tip of Mount Everest (5.5 miles/8.85 kilometers), drif
18、ting even higher than the cruising altitude of commercial airliners (5.6 miles/9.17 kilometers) and into the stratosphere. As he crosses the boundary layer (called the tropopause), he can expect a lot of turbulence.The balloon will slowly drift to the edge of space at 120,000 feet (22.7 miles/36.53
19、kilometers). Here, Fearless Felix will unclip. He will roll back the door.Then, I would assume, he will slowly step out onto something resembling an Olympic diving platform.Below, the Earth becomes the concrete bottom of a swimming pool that he wants to land on, but not too hard. Still, hell be trav
20、eling fast, so despite the distance, it will not be like diving into the deep end of a pool. It will be like he is diving into the shallow end.Skydiver preps for the big jumpWhen he jumps, he is expected to reach the speed of sound - 690 mph (1,110 kph) - in less than 40 seconds. Like hitting the to
21、p of the water, he will begin to slow as he approaches the more dense air closer to Earth. But this will not be enough to stop him completely.If he goes too fast or spins out of control, he has a stabilization parachute that can be deployed to slow him down. His team hopes its not needed. Instead, h
22、e plans to deploy his 270-square-foot (25-square-meter) main chute at an altitude of around 5,000 feet (1,524 meters).In order to deploy this chute successfully, he will have to slow to 172 mph (277 kph). He will have a reserve parachute that will open automatically if he loses consciousness at mach
23、 speeds.Even if everything goes as planned, it wont. Baumgartner still will free fall at a speed that would cause you and me to pass out, and no parachute is guaranteed to work higher than 25,000 feet (7,620 meters).It might not be the moon, but Kittinger free fell from 102,800 feet in 1960 - at the dawn of an infamous space race that captured the hearts of many. Baumgartner will attempt to break that record, a feat that boggles the mind. This is one of those monumental moments I will always remember, because there is no way Id miss this.
