Sequences and Series (page 1 of 5)
Sections: Terminology and notation, Basic examples, Arithmetic and geometric sequences, Arithmetic series, Finite and infinite geometric series
Maths in a minute: Pretend primes
Submitted by mf344 on July 8, 2015
Suppose you have a prime number $p$ and some other natural number $x$. Then, no matter what the value of $x$ is, as long as it’s a natural number, you will find that
[ x^ p - x ]
is a multiple of $p.$
This result is known as Fermat's little theorem, not to be confused with Fermat's last theorem.
Let’s try the little theorem with a few examples. For $p=2$ and $x=5$ we have
[ 5^2 - 5 = 25 -5 = 20 = 10 imes 2. ]
For $p = 3$ and $x = 2$ we have
[ 2^3 - 2 = 8 - 2 = 6 = 2 imes 3. ]
And for $p = 7$ and $x =11$ we have
[ 11^7 - 11 = 19,487,171 - 11 = 19,487,160 = 2,783,880 imes 7. ]
You can try it out for other values of $p$ and $x$ yourself.
Fermat
Pierre de Fermat.
Fermat first mentioned a version of this theorem in a letter in 1640. As with his last theorem, he was a little cryptic about the proof:
"...the proof of which I would send to you, if I were not afraid to be too long."
But unlike with Fermat's last theorem, a proof was published relatively soon; in 1736 by Leonhard Euler.
But does Fermat’s little theorem work the other way around? If you have a natural number $n$ so that for all other natural numbers $x$
$x^ n - x$
is a multiple of $n,$ does this imply that $n$ is a prime?
If this were true, then we could use Fermat’s little theorem to check whether a given number $n$ is prime: pick a bunch of other numbers $x$ at random, and for each of them check whether
[ x^ n - x ]
is a multiple of $n.$ If you find an $x$ for which this isn’t true, then you know for sure that $n$ isn’t prime. If you don’t find one, then provided you have checked sufficiently many $x,$ you can be pretty sure that $n$ is prime. This method is called Fermat’s primality test.
Alas, it doesn’t quite work as well as it could. In 1885 the Czech mathematician Václav Šimerka discovered non-prime numbers that masquerade as primes when it comes to Fermat’s little theorem. The number $561$ is the smallest of them. It’s not prime, but for all other natural numbers $x$ we have that
[ x^{561} - x ]
is a multiple of $561.$
Šimerka also discovered that $1105, 1729, 2465, 2821, 6601$ and $8911$ behave in the same way. Natural numbers that aren't primes but satisfy the relationship stated in Fermat's little theorem are sometimes called pseudoprimes, because they make such a good job of behaving like primes, or Carmichael numbers, after the American Robert Carmichael, who independently found the first one, 561, in 1910.
You can see from the first seven named above that Carmichael numbers aren’t too abundant. There are infinitely many of them, a fact that wasn’t proved until 1994, but they are very sparse. In fact, they get sparser as you move up the number line: if you count the Carmichael numbers between 1 and $10^{21}$, you’ll find that there are only around one in 50 trillion.
Carmichael numbers do hamper Fermat's primality test somewhat, but not so badly as to make it totally unusable. And there are modified versions of the test that work very well. As cans of worms opened by Fermat go, the one involving Carmichael numbers definitely wasn't the worst.
A "sequence" (or "progression", in British English) is an ordered list of numbers; the numbers in this ordered list are called "elements" or "terms". A "series" is the value you get when you add up all the terms of a sequence; this value is called the "sum". For instance, "1, 2, 3, 4" is a sequence, with terms "1", "2", "3", and "4"; the corresponding series is the sum "1 + 2 + 3 + 4", and the value of the series is 10.
A sequence may be named or referred to as "A" or "An". The terms of a sequence are usually named something like "ai" or "an", with the subscripted letter "i" or "n" being the "index" or counter. So the second term of a sequnce might be named "a2" (pronounced "ay-sub-two"), and "a12" would designate the twelfth term.
Note: Sometimes sequences start with an index of n = 0, so the first term is actually a0. Then the second term would be a1. The first listed term in such a case would be called the "zero-eth" term. This method of numbering the terms is used, for example, in Javascript arrays. Don't assume that every sequence and series will start with an index of n = 1.
A sequence A with terms an may also be referred to as "{an}", but contrary to what you may have learned in other contexts, this "set" is actually an ordered list, not an unordered collection of elements. (Your book may use some notation other than what I'm showing here. Unfortunately, notation doesn't yet seem to have been entirely standardized for this topic. Just try always to make sure, whatever resource you're using, that you are clear on the definitions of that resource's terms and symbols.)
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To indicate a series, we use either the Latin capital letter "S" or else the Greek letter corresponding to the capital "S", which is called "sigma" (SIGG-muh):
sigma
To show the summation of, say, the first through tenth terms of a sequence {an}, we would write the following:
sigma; "n = 1" below, "10" above, and "a-sub-n" to the right
The "n = 1" is the "lower index", telling us that "n" is the counter and that the counter starts at "1"; the "10" is the "upper index", telling us that a10 will be the last term added in this series; "an" stands for the terms that we'll be adding. The whole thing is pronounced as "the sum, from n equals one to ten, of a-sub-n". The summation symbol above means the following:
a1 + a2 + a3 + a4 + a5 + a6 + a7 + a8 + a9 + a10
The written-out form above is called the "expanded" form of the series, in contrast with the more compact "sigma" notation. Copyright © Elizabeth Stapel 2006-2011 All Rights Reserved
Any letter can be used for the index, but i, j, k, and n are probably used more than any other letters.
Sequences and series are most useful when there is a formula for their terms. For instance, if the formula for an is "2n + 3", then you can find the value of any term by plugging the value of n into the formula. For instance, a8 = 2(8) + 3 = 16 + 3 = 19. In words, "an = 2n + 3" can be read as "the n-th term is given by two-enn plus three". The word "n-th" is pronounced "ENN-eth", and just means "the generic term an, where I haven't yet specified the value of n."
Of course, there doesn't have to be a formula for the n-th term of a sequence. The values of the terms can be utterly random, having no relationship between n and the value of an. But sequences with random terms are hard to work with and are less useful in general, so you're not likely to see many of them in your classes.
Sequences and Series (page 1 of 5)Sections: Terminology and notation, Basic examples, Arithmetic and geometric sequences, Arithmetic series, Finite and infinite geometric seriesMaths in a minute: Pretend primesSubmitted by mf344 on July 8, 2015Suppose you have a prime number $p$ and some other natural number $x$. Then, no matter what the value of $x$ is, as long as it’s a natural number, you will find that [ x^ p - x ] is a multiple of $p.$This result is known as Fermat's little theorem, not to be confused with Fermat's last theorem.Let’s try the little theorem with a few examples. For $p=2$ and $x=5$ we have [ 5^2 - 5 = 25 -5 = 20 = 10 imes 2. ] For $p = 3$ and $x = 2$ we have [ 2^3 - 2 = 8 - 2 = 6 = 2 imes 3. ] And for $p = 7$ and $x =11$ we have [ 11^7 - 11 = 19,487,171 - 11 = 19,487,160 = 2,783,880 imes 7. ] You can try it out for other values of $p$ and $x$ yourself.FermatPierre de Fermat.Fermat first mentioned a version of this theorem in a letter in 1640. As with his last theorem, he was a little cryptic about the proof:"...the proof of which I would send to you, if I were not afraid to be too long."But unlike with Fermat's last theorem, a proof was published relatively soon; in 1736 by Leonhard Euler.But does Fermat’s little theorem work the other way around? If you have a natural number $n$ so that for all other natural numbers $x$$x^ n - x$is a multiple of $n,$ does this imply that $n$ is a prime?If this were true, then we could use Fermat’s little theorem to check whether a given number $n$ is prime: pick a bunch of other numbers $x$ at random, and for each of them check whether [ x^ n - x ] is a multiple of $n.$ If you find an $x$ for which this isn’t true, then you know for sure that $n$ isn’t prime. If you don’t find one, then provided you have checked sufficiently many $x,$ you can be pretty sure that $n$ is prime. This method is called Fermat’s primality test.Alas, it doesn’t quite work as well as it could. In 1885 the Czech mathematician Václav Šimerka discovered non-prime numbers that masquerade as primes when it comes to Fermat’s little theorem. The number $561$ is the smallest of them. It’s not prime, but for all other natural numbers $x$ we have that [ x^{561} - x ] is a multiple of $561.$Šimerka also discovered that $1105, 1729, 2465, 2821, 6601$ and $8911$ behave in the same way. Natural numbers that aren't primes but satisfy the relationship stated in Fermat's little theorem are sometimes called pseudoprimes, because they make such a good job of behaving like primes, or Carmichael numbers, after the American Robert Carmichael, who independently found the first one, 561, in 1910.You can see from the first seven named above that Carmichael numbers aren’t too abundant. There are infinitely many of them, a fact that wasn’t proved until 1994, but they are very sparse. In fact, they get sparser as you move up the number line: if you count the Carmichael numbers between 1 and $10^{21}$, you’ll find that there are only around one in 50 trillion.Carmichael numbers do hamper Fermat's primality test somewhat, but not so badly as to make it totally unusable. And there are modified versions of the test that work very well. As cans of worms opened by Fermat go, the one involving Carmichael numbers definitely wasn't the worst.A "sequence" (or "progression", in British English) is an ordered list of numbers; the numbers in this ordered list are called "elements" or "terms". A "series" is the value you get when you add up all the terms of a sequence; this value is called the "sum". For instance, "1, 2, 3, 4" is a sequence, with terms "1", "2", "3", and "4"; the corresponding series is the sum "1 + 2 + 3 + 4", and the value of the series is 10.A sequence may be named or referred to as "A" or "An". The terms of a sequence are usually named something like "ai" or "an", with the subscripted letter "i" or "n" being the "index" or counter. So the second term of a sequnce might be named "a2" (pronounced "ay-sub-two"), and "a12" would designate the twelfth term.Note: Sometimes sequences start with an index of n = 0, so the first term is actually a0. Then the second term would be a1. The first listed term in such a case would be called the "zero-eth" term. This method of numbering the terms is used, for example, in Javascript arrays. Don't assume that every sequence and series will start with an index of n = 1.
A sequence A with terms an may also be referred to as "{an}", but contrary to what you may have learned in other contexts, this "set" is actually an ordered list, not an unordered collection of elements. (Your book may use some notation other than what I'm showing here. Unfortunately, notation doesn't yet seem to have been entirely standardized for this topic. Just try always to make sure, whatever resource you're using, that you are clear on the definitions of that resource's terms and symbols.)
ADVERTISEMENT
To indicate a series, we use either the Latin capital letter "S" or else the Greek letter corresponding to the capital "S", which is called "sigma" (SIGG-muh):
sigma
To show the summation of, say, the first through tenth terms of a sequence {an}, we would write the following:
sigma; "n = 1" below, "10" above, and "a-sub-n" to the right
The "n = 1" is the "lower index", telling us that "n" is the counter and that the counter starts at "1"; the "10" is the "upper index", telling us that a10 will be the last term added in this series; "an" stands for the terms that we'll be adding. The whole thing is pronounced as "the sum, from n equals one to ten, of a-sub-n". The summation symbol above means the following:
a1 + a2 + a3 + a4 + a5 + a6 + a7 + a8 + a9 + a10
The written-out form above is called the "expanded" form of the series, in contrast with the more compact "sigma" notation. Copyright © Elizabeth Stapel 2006-2011 All Rights Reserved
Any letter can be used for the index, but i, j, k, and n are probably used more than any other letters.
Sequences and series are most useful when there is a formula for their terms. For instance, if the formula for an is "2n + 3", then you can find the value of any term by plugging the value of n into the formula. For instance, a8 = 2(8) + 3 = 16 + 3 = 19. In words, "an = 2n + 3" can be read as "the n-th term is given by two-enn plus three". The word "n-th" is pronounced "ENN-eth", and just means "the generic term an, where I haven't yet specified the value of n."
Of course, there doesn't have to be a formula for the n-th term of a sequence. The values of the terms can be utterly random, having no relationship between n and the value of an. But sequences with random terms are hard to work with and are less useful in general, so you're not likely to see many of them in your classes.
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