561
Christmas break reading included Those Fascinating Numbers
by Jean-Marie De Konnick. The plot is simply one to the Skewes number.
Many of the concepts are simply beyond me due to my own lack of grounding in number theory. I was, however, introduced to Fermat's Little Theorem and became interested in the Carmichael numbers. The smallest Carmichael number, however, is 561. An attempt, however, to calculate the number 2 560 on a calculator, let alone any other base, caused smoke to come out of my calculator. An error message was all I could achieve.
Still desiring to see if indeed 561 divided evenly into ((2^560)-1), I decided to throw the equation at WolframAlpha. I entered ((2^560)-1)/561 and WolframAlpha dutifully returned 6727205748344993497756781784292814422311802531936613954324788413985069808185648577693630526163384612373383912710808992691213163333199273854833216134461133767645980975 and no remainder. Little wonder my calculator failed to get that right. 166 digits. Impressive.
That result being rather messy, I went ahead and used the modulo definition of the Fermat's Little Theorem to obtain a "prettier" result. Entering (2^560) mod 561 yielded a result of one.
Along the way I also came to appreciate WolframAlpha's ability to report prime factors and, even better, all divisors. Thus factor 13665960 yields the 10 prime factors, 7 distinct, and all 384 divisors. Working out the number of divisors by hand is simply error prone beyond n = 100.
Why would one ever again do any mathematics without WolframAlpha? What is the benefit to the drill and kill working out of equations? Mathematicians tend to say, "But if they do not do it by hand, they will not understand it." OK, so before you get to drive a car from Miami to Chicago, you need to walk the route first so you can understand it. Why not let the computer handle the drudgery, and use tools like WolframAlpha to explore mathematics in ways that simply were not previously possible?
Of course, then every mathematics student would have to have a laptop and a wireless Internet connection. But why not? Who is ever going to try to calculate 2^560 by hand? Yet by the same token, why worry about hand factoring quadratics? Both are drudgery, the former is simply more so. Move beyond drudgery and make mathematics the exciting exploration of terra nova that it can be.
by Jean-Marie De Konnick. The plot is simply one to the Skewes number.Many of the concepts are simply beyond me due to my own lack of grounding in number theory. I was, however, introduced to Fermat's Little Theorem and became interested in the Carmichael numbers. The smallest Carmichael number, however, is 561. An attempt, however, to calculate the number 2 560 on a calculator, let alone any other base, caused smoke to come out of my calculator. An error message was all I could achieve.
Still desiring to see if indeed 561 divided evenly into ((2^560)-1), I decided to throw the equation at WolframAlpha. I entered ((2^560)-1)/561 and WolframAlpha dutifully returned 6727205748344993497756781784292814422311802531936613954324788413985069808185648577693630526163384612373383912710808992691213163333199273854833216134461133767645980975 and no remainder. Little wonder my calculator failed to get that right. 166 digits. Impressive.
That result being rather messy, I went ahead and used the modulo definition of the Fermat's Little Theorem to obtain a "prettier" result. Entering (2^560) mod 561 yielded a result of one.
Along the way I also came to appreciate WolframAlpha's ability to report prime factors and, even better, all divisors. Thus factor 13665960 yields the 10 prime factors, 7 distinct, and all 384 divisors. Working out the number of divisors by hand is simply error prone beyond n = 100.
Why would one ever again do any mathematics without WolframAlpha? What is the benefit to the drill and kill working out of equations? Mathematicians tend to say, "But if they do not do it by hand, they will not understand it." OK, so before you get to drive a car from Miami to Chicago, you need to walk the route first so you can understand it. Why not let the computer handle the drudgery, and use tools like WolframAlpha to explore mathematics in ways that simply were not previously possible?
Of course, then every mathematics student would have to have a laptop and a wireless Internet connection. But why not? Who is ever going to try to calculate 2^560 by hand? Yet by the same token, why worry about hand factoring quadratics? Both are drudgery, the former is simply more so. Move beyond drudgery and make mathematics the exciting exploration of terra nova that it can be.
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