tag:blogger.com,1999:blog-5303307482158922565.post8950612277953560220..comments2024-03-22T13:39:55.941-07:00Comments on Math Mama Writes...: Tutoring & Number TheorySue VanHattumhttp://www.blogger.com/profile/10237941346154683902noreply@blogger.comBlogger4125tag:blogger.com,1999:blog-5303307482158922565.post-50849972854368370532011-03-03T21:51:02.938-08:002011-03-03T21:51:02.938-08:00Everyone's being so good about hiding their co...Everyone's being so good about hiding their complete thoughts. :^) I guess I'll say a little here. Odd numbers were easy for me. I showed 'Artemis' a proof as I thought it out. Even numbers felt more confusing, but then I thought about it in bed, and realized I had it. <br /><br />If you have it, you know which even numbers work.<br /><br />Ben, I'll have to think about Wilson's Theorem when I'm not so tired.Sue VanHattumhttps://www.blogger.com/profile/10237941346154683902noreply@blogger.comtag:blogger.com,1999:blog-5303307482158922565.post-4853872962766770022011-03-03T13:53:38.610-08:002011-03-03T13:53:38.610-08:00Yes, fab problem. I was going to say I hadn't ...Yes, fab problem. I was going to say I hadn't figured it out yet but as I started to write down my approach it became clear to me that I had. So, for n=732 the sum does not divide the factorial, but for n=733 it does... is this what you're getting Avery and Blaise?<br /><br />My approach, like the others, began with writing a closed form for the sum. Then I multiplied and divided some stuff and reduced everything mod n+1. Then I realized I knew what was going on and it reminded me of <a href="http://en.wikipedia.org/wiki/Wilson%27s_theorem" rel="nofollow">Wilson's Theorem</a>.Anonymousnoreply@blogger.comtag:blogger.com,1999:blog-5303307482158922565.post-68272819856453608492011-03-03T12:55:22.592-08:002011-03-03T12:55:22.592-08:00I too considered a closed expression for the sum f...I too considered a closed expression for the sum from 1 to n and thought I had an answer pretty quickly. Realized from Blaise's comment that I was missing a piece. I now think that I've dealt with this 2nd part and have an informal proof. So if n=100, the sum isn't a divisor of the factorial but the sum is a divisor when n=101.Avery Pickfordhttps://www.blogger.com/profile/10433339146333801163noreply@blogger.comtag:blogger.com,1999:blog-5303307482158922565.post-1990374354591548312011-03-02T18:19:43.699-08:002011-03-02T18:19:43.699-08:00Cute problem. I looked at the formulas for comput...Cute problem. I looked at the formulas for computing factorials and triangular numbers and sought insight. Once I had an idea, I then through some carefully chosen test cases at it to validate the idea.<br /><br />For instance, I verified that when n=36, the sum is not a divisor of the product, but it is when n=38.<br /><br />I am confident of my solution, but I haven't proven it.Buddha Buckhttps://www.blogger.com/profile/17167036913705912859noreply@blogger.com