2.7: Theorems and Conjectures involving prime numbers

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We have proved that there are infinitely many primes. The question that arises naturally here is the following: Can we estimate how many primes are there less than a given number? The theorem that answers this question is the prime number theorem. We denote by (pi(x)) the number of primes less than a given positive number (x). Many mathematicians worked on this theorem and conjectured many estimates before Chebyshev finally stated that the estimate is (x/log x). The prime number theorem was finally proved in 1896 when Hadamard and Poussin produced independent proofs. Before stating the prime number theorem, we state and prove a lemma involving primes that will be used in the coming chapters.

Lemma

Let (p) be a prime and let (min mathbb{Z^+}). Then the highest power of (p) dividing (m!) is [sum_{i=1}^inftyleft[frac{m}{p^i} ight]]

Among all the integers from 1 till (m), there are exactly (left[frac{m}{p} ight]) integers that are divisible by (p). These are (p,2p,...,left[frac{m}{p} ight]p). Similarly we see that there are (left[frac{m}{p^i} ight]) integers that are divisible by (p^i). As a result, the highest power of (p) dividing (m!) is

[sum_{igeq 1}ileft{left[frac{m}{p^i} ight]-left[frac{m}{p^{i+1}} ight] ight}=sum_{igeq 1} left[frac{m}{p^i} ight]]

The Prime Number Theorem

Let (x>0) then [pi(x)sim x/log x]

So this theorem says that you do not need to find all the primes less than (x) to find out their number, it will be enough to evaluate (x/log x) for large (x) to find an estimate for the number of primes. Notice that I mentioned that (x) has to be large enough to be able to use this estimate.

Several other theorems were proved concerning prime numbers. many great mathematicians approached problems that are related to primes. There are still many open problems of which we will mention some.

Twin Prime Conjecture

There are infinitely many pairs primes (p) and (p+2).

Goldbach’s Conjecture

Every even positive integer greater than 2 can be written as the sum of two primes.

The (n^2+1) Conjecture

There are infinitely many primes of the form (n^2+1), where (n) is a positive integer.

Polignac Conjecture

For every even number (2n) are there infinitely many pairs of consecutive primes which differ by (2n).

Opperman Conjecture

Is there always a prime between (n^2) and ((n+1)^2)?

I second Martin's recommendation of Pomerance & Crandall.

On the popularizer level we have books like George P. Loweke's The Lore of Prime Numbers and David Wells's Prime Numbers: The Most Mysterious Figures in Math.

Somewhere in the middle is Ribenboim's Little Book of Bigger Primes.

On a more advanced level there are books like Fine & Rosenberger's Number Theory: An Introduction Via the Distribution of Primes and David Cox's Primes of the Form $x^2 + ny^2$: From Fermat, Class Field Theory, and Complex Multiplication. That one might be a little difficult to search for in your library's computerized catalog, plus it assumes a lot of knowledge of advanced algebra.

By the way, these are all books I have checked out from a library at one time or another. If I were you, I'd just casually browse in the vicinity of QA 240 in your university's library and 510 in your public library.

Assuming Polignac's conjecture we will always be able to find two primes $(c,d)$ such that

(the distance between $a$ and $b$ along the $x$-axis is equal to the negative of the distance between $c$ and $d$) and

($a$ and $b$, and $c$ and $d$, end in the same digits).

This defines an isosceles trapezium, which is always a cyclic quadrilateral (a quadrilateral such that a circle can be drawn with its 4 vertices.

If $a = b mod 10$, the above argument still probably holds, but I have not found a proof.

Here is some loose intuition to convince you that it is equally hard as the twin prime conjecture. Especially, to convince you that there's no point in trying to prove or disprove it:

At most as hard as the twin prime conjecture:
Take two primes $p_1,p_2$. If the twin prime conjecture is true, it is reasonable to expect that, for any even $2k geq 2$ and $n mod 10$ there are infinitely many prime pairs $(q_1,q_2)$ with $q_2-q_1 = 2k$ and $q_1 equiv n pmod<10>$. 1 Then for any given $p_1,p_2$ not congruent mod $10$ we can find two other primes to form a trapezium. This takes care of the case where $p_1,p_2$ are not congruent, at least.

At least as hard as the twin prime conjecture:
Four points with coordinates $(x_i,y_i)$ are concyclic iff $egin 1 & x_1 & y_1 & x_1^2 + y_1^2 1 & x_2 & y_2 & x_2^2 + y_2^2 1 & x_3 & y_3 & x_3^2 + y_3^2 1 & x_4 & y_4 & x_4^2 + y_4^2 end = 0$ This gives, for every pair of primes $(p_1,p_2)$ a degree $4$ equation in two primes $q_1,q_2$ (and their residues mod $10$). Current methods are nowhere near proving that it has a solution indeed, we cannot even show that the degree 1 (!) equation $q_2-q_1-2k = 0$ has a solution for every $k$.

1 Although, there was an article that appeared a few years ago with some computations, suggesting that the distribution of the remainder of three consecutive primes mod a given integer, is not uniform. Anyway.

Goldbach’s Conjecture

This is another simply stated problem. Goldbach’s Conjecture says that every even number larger than two can be written as the sum of two prime numbers. This certainly holds true for smaller numbers: 4 = 2 + 2, 8 = 5 + 3, 20 = 13 + 7, but it hasn’t been proven for all even numbers.

Researchers armed with 21st century computers and well-designed programs have verified the conjecture for even numbers up to 4,000,000,000,000,000,000. This is pretty good evidence for the conjecture, but in mathematics, saying that a conjecture holds for all numbers smaller than some ludicrously high finite bound is not enough to say that it holds for all numbers.

MAT 112 Ancient and Contemporary Mathematics

It was relatively easy to prove that there are infinitely many primes (Theorem 10.4.1). In order to come up with a new mathematical result, a great deal of study, investigation, and insight is often required. Ideas arise, steps toward a proof are taken, and sometimes those ideas have to be tweaked. In this process, it is possible to develop a statement that is believed to be true but has not been formally proven. Such a statement is called a conjecture and is often known in mathematics as an open problem. We conclude this section by presenting an important conjecture involving primes. While the statement of the conjecture is easy to understand and computer experiments have not come up with a counterexample, we do not know whether it is true.

We give an overview over the twin prime conjecture in the video in Figure 10.5.1 Further details are given below.

Example 10.5.3 . Twin primes.

The first four twin prime pairs are

Problem 10.5.4 . Recognize twin primes.

Determine whether or not (89) is a part of a twin prime pair.

Problem 10.5.5 . Recognize twin primes.

Determine whether or not (137) is a part of a twin prime pair.

The Twin Prime Conjecture is the claim that there are infinitely many twin prime pairs.

Conjecture 10.5.6 . Twin Prime Conjecture.

There are infinitely many primes (p) such that (p + 2) is also prime.

This is the first (and only) conjecture that you will encounter in this course. It is important to distinguish conjectures and theorems. Both conjectures and theorems are statements. While theorems are true statements, for a conjecture nobody has determined yet whether it is true or false. As soon as it is determined by a proof that a conjecture is true, it becomes a theorem. Also see the treatment of this topic in the preface in Subsection 4.

Checkpoint 10.5.7 . Are these conjectures, definitions, or theorems ?

It is outside the scope of this course to try to prove the twin prime conjecture. Nevertheless it is interesting to see whether twin primes exist (if not the conjecture would be false and not of much interest).

Problem 10.5.8 . Count twin prime pairs.

How many twin prime pairs are there up to (100) ?

With Table 10.2.4 we get that the twin prime pairs up to (100) are:

It appears that there are fewer twin primes than there are primes. In Checkpoint 10.5.9 count the number of primes and twin primes up to a given natural number.

Checkpoint 10.5.9 . Count primes and twin prime pairs.

Progress towards proving the twin prime conjecture (Conjecture 10.5.6) has been made recently. In 2013, Yitang Zhang [10] made a major breakthrough by proving that there are infinitely many primes (p) and (q) such that (p-qle 70,000,000 ext<.>)

Soon after this was improved considerably, such that now it is known that there are infinitely many primes (p) and (q) such that (p-qle 246 ext<.>) When it is proven that there are infinitely many primes (p) and (q) such that (p-qle 2 ext<,>) the twin prime conjecture is proven.

We end this section with a song about the twin prime conjecture in Figure 10.5.10.

"Like a Shakespearean sonnet that captures the very essence of love, or a painting that brings out the beauty of the human form that is far more than just skin deep, Euler's Equation reaches down into the very depths of existence."

Stanford mathematician Keith Devlin wrote these words about the equation to the left in a 2002 essay called "The Most Beautiful Equation." But why is Euler's formula so breath-taking? And what does it even mean?

First, the letter "e" represents an irrational number (with unending digits) that begins 2.71828. Discovered in the context of continuously compounded interest, it governs the rate of exponential growth, from that of insect populations to the accumulation of interest to radioactive decay. In math, the number exhibits some very surprising properties, such as &mdash to use math terminology &mdash being equal to the sum of the inverse of all factorials from 0 to infinity. Indeed, the constant "e" pervades math, appearing seemingly from nowhere in a vast number of important equations.

Next, "i" represents the so-called "imaginary number": the square root of negative 1. It is thus called because, in reality, there is no number which can be multiplied by itself to produce a negative number (and so negative numbers have no real square roots). But in math, there are many situations where one is forced to take the square root of a negative. The letter "i" is therefore used as a sort of stand-in to mark places where this was done.

Pi, the ratio of a circle's circumference to its diameter, is one of the best-loved and most interesting numbers in math. Like "e," it seems to suddenly arise in a huge number of math and physics formulas. What Makes Pi So Special?]

Putting it all together, the constant "e" raised to the power of the imaginary "i" multiplied by pi equals -1. And, as seen in Euler's equation, adding 1 to that gives 0. It seems almost unbelievable that all these strange numbers &mdash and even one that isn't real &mdash would combine so simply. But it's a proven fact.

What are prime numbers, and why are they so vital to modern life?

If you’ve graduated high-school and you’re reading this article, you probably at least know the following about prime numbers: Primes are the set of all numbers that can only be equally divided by 1 and themselves, with no other even division possible. Numbers like 2, 3, 5, 7, and 11 are all prime numbers. What fewer people know is why these numbers are so important, and how the mathematical logic behind them has resulted in vital applications in the modern world.

Here’s something cool about primes: Mathematicians have shown that absolutely any whole number can be expressed as a product of primes, only primes, and nothing else. For example:

To get 222, try 2 * 3 * 37

123,228,940? Why, that’s just, 2 * 2 * 5 * 23 * 79 * 3391

This rule, called the prime factorization rule, is called something else as well: the Fundamental Theorem of Arithmetic. It makes sense when we think about what primes are, numbers that can’t be pulled apart any further. So as we try to pull apart any number into two numbers, then pull those apart into two numbers if possible, and so on, we will eventually be left only with primes.

This all might seem like nothing more than a cool mathematical oddity. But it becomes important thanks to one simple additional fact: As far as the best mathematicians and computer scientists have been able to determine, it is totally impossible to come up with a truly efficient formula for factoring large numbers into primes.

That is to say, we have ways of factoring large numbers into primes, but if we try to do it with a 200-digit number, or a 500-digit number, using the same algorithms we would use to factor a 7-digit number, the world’s most advanced supercomputers still take absurd amounts of time to finish. Like, timescales longer than the formation of the planet and, for extremely large numbers, longer than the age of the universe itself.

So, there is a functional limit to the size of the numbers we can factor into primes, and this fact is absolutely essential to modern computer security. Pretty much anything that computers can easily do without being able to easily undo will be of interest to computer security. Modern encryption algorithms exploit the fact that we can easily take two large primes and multiply them together to get a new, super-large number, but that no computer yet created can take that super-large number and quickly figure out which two primes went into making it.

This math-level security allows what’s called public key cryptography, or encryption where we don’t have to worry about publishing a key to use in encrypting transmissions, because simply having that key (a very large number) won’t help anyone to undo the encryption it created. In order to undo the encryption, and read the message, you need the prime factors of the key used for encryption — and as we’ve been seeing, that’s not something you can just figure out on your own.

This allows us to get around the core paradox of encryption: How do you securely communicate the initial specifics needed to set up secure communication in the first place? In public key cryptography, which is the backbone of computer encryption, we can get around this because the specifics of how to get into secure contact don’t themselves need to be secure. Quite the opposite — people generally post links to their public keys on social media, so as many people as possible will be able to encrypt messages for them. Though there are now quite a few encryption algorithms that exploit prime factorization, the most historically significant, and still the conceptual blueprint for the field, is called RSA.

Whether it’s communicating your credit card information to Amazon, logging into your bank, or sending a manually encrypted email to a colleague, we are constantly using computer encryption. And that means we are constantly using prime numbers, and relying on their odd numerical properties for protection of the cyber-age way of life. It’s no meaningless academic quest, the effort to better understand prime numbers, since virtually all of modern security relies upon the current limitations of that understanding.

That’s all not to say that there has been no progress in factoring large numbers. In 2009, researchers networked several hundred computers together and spent the equivalent of about 2,000 years for a single computer, using advanced factoring algorithms to factor the “RSA-768” number — that is to say, a number with 232 digits put up by the RSA group as a factoring challenge. Proving it was possible to break 768-bit encryption in non-universal-heat-death timescales is unacceptable for the security world, of course, and so the standard has now moved to RSA-1024, using numbers with 309 digits.

1024-bit encryption ought to still be safe from anyone not in possession of a time machine, so far as we know — though rumors abound on the internet of secret quantum computer projects at the NSA or elsewhere, ones that can chew through even 2048-bit encryption like it ain’t nothing. There’s absolutely no evidence that such a thing exists, however.

Prime numbers are cool. As Carl Sagan points out so eloquently in the novel Contact, there’s a certain importance to their status as the most fundamental building block of all numbers, which are themselves the building blocks of our understanding of the universe. In that book, aliens choose to send a long string of prime numbers as proof that their message is intelligent and not natural in origin, since primes are one thing that cannot change due to differences of psychology, lifestyle, or evolutionary history. No matter what an advanced alien life-form looks or thinks like, if it understands the world around it, it almost certainly has the concept of a prime.

That’s why a lot of mathematicians view number theory as a little bit like archaeology. The feeling isn’t one of inventing new technologies, but of uncovering the logical foundations of the universe, those that describe its behavior everywhere, throughout all of time.

Check out our ExtremeTech Explains series for more in-depth coverage of today’s hottest tech topics.

2.7: Theorems and Conjectures involving prime numbers

In addition to being a Topcoder member, medv is a lecturer in Kiev National University’s cybernetics faculty.

Prime numbers and their properties were extensively studied by the ancient Greek mathematicians. Thousands of years later, we commonly use the different properties of integers that they discovered to solve problems. In this article we’ll review some definitions, well-known theorems, and number properties, and look at some problems associated with them.

A prime number is a positive integer, which is divisible on 1 and itself. The other integers, greater than 1, are composite. Coprime integers are a set of integers that have no common divisor other than 1 or -1.

The fundamental theorem of arithmetic:
Any positive integer can be divided in primes in essentially only one way. The phrase ‘essentially one way’ means that we do not consider the order of the factors important.

One is neither a prime nor composite number. One is not composite because it doesn’t have two distinct divisors. If one is prime, then number 6, for example, has two different representations as a product of prime numbers: 6 = 2 * 3 and 6 = 1 * 2 * 3. This would contradict the fundamental theorem of arithmetic.

Euclid’s theorem:
There is no largest prime number.

To prove this, let’s consider only n prime numbers: p1, p2, …, pn. But no prime pi divides the number

so N cannot be composite. This contradicts the fact that the set of primes is finite.

Exercise 1. Sequence an is defined recursively:

Prove that ai and aj, i ¹ j are relatively prime.

Hint: Prove that an+1 = a1a2…an + 1 and use Euclid’s theorem.

Exercise 2. Ferma numbers Fn (n ≥ 0) are positive integers of the form

Prove that Fi and Fj, i ≠ j are relatively prime.

Hint: Prove that Fn +1 = F0F1F2…Fn + 2 and use Euclid’s theorem.

For any two positive coprime integers a and b there are infinitely many primes of the form a + n*b, where n > 0.

Trial division:
Trial division is the simplest of all factorization techniques. It represents a brute-force method, in which we are trying to divide n by every number i not greater than the square root of n. (Why don’t we need to test values larger than the square root of n?) The procedure factor prints the factorization of number n. The factors will be printed in a line, separated with one space. The number n can contain no more than one factor, greater than n.

Consider a problem that asks you to find the factorization of integer g(-231 < g <231) in the form

g = f1 x f2 x … x fn or g = -1 x f1 x f2 x … x fn

where fi is a prime greater than 1 and fi ≤ fj for i < j.

For example, for g = -192 the answer is -192 = -1 x 2 x 2 x 2 x 2 x 2 x 2 x 3.

To solve the problem, it is enough to use trial division as shown in function factor.

Sieve of Eratosthenes:
The most efficient way to find all small primes was proposed by the Greek mathematician Eratosthenes. His idea was to make a list of positive integers not greater than n and sequentially strike out the multiples of primes less than or equal to the square root of n. After this procedure only primes are left in the list.

The procedure of finding prime numbers gen_primes will use an array primes[MAX] as a list of integers. The elements of this array will be filled so that

At the beginning we mark all numbers as prime. Then for each prime number i (i ≥ 2), not greater than √MAX, we mark all numbers ii, i(i + 1), … as composite.

For example, if MAX = 16, then after calling gen_primes, the array ‘primes’ will contain next values:

i0123456789101112131415
primes[i]1111010100010100

Goldbach’s Conjecture:
For any integer n (n ≥ 4) there exist two prime numbers p1 and p2 such that p1 + p2 = n. In a problem we might need to find the number of essentially different pairs (p1, p2), satisfying the condition in the conjecture for a given even number n (4 ≤ n ≤ 2 15). (The word ‘essentially’ means that for each pair (p1, p2) we have p1 ≤p2.)

For example, for n = 10 we have two such pairs: 10 = 5 + 5 and 10 = 3 + 7.

To solve this,as n ≤ 215 = 32768, we’ll fill an array primes[32768] using function gen_primes. We are interested in primes, not greater than 32768.

The function FindSol(n) finds the number of different pairs (p1, p2), for which n = p1 + p2. As p1 ≤ p2, we have p1 ≤ n/2. So to solve the problem we need to find the number of pairs (i, n – i), such that i and n – i are prime numbers and 2 ≤ i ≤ n/2.

Euler’s totient function
The number of positive integers, not greater than n, and relatively prime with n, equals to Euler’s totient function φ (n). In symbols we can state that

This function has the following properties:

If p is prime, then φ § = p – 1 and φ (pa) = p a * (1 – 1/p) for any a.
If m and n are coprime, then φ (m * n) = φ (m) * φ (n).
If n = , then Euler function can be found using formula:
φ (n) = n * (1 – 1/p 1) * (1 – 1/p 2) * … * (1 – 1/p k)

The function fi(n) finds the value of φ(n):

For example, to find φ(616) we need to factorize the argument: 616 = 23 * 7 * 11. Then, using the formula, we’ll get:

φ(616) = 616 * (1 – 1/2) * (1 – 1/7) * (1 – 1/11) = 616 * 1/2 * 6/7 * 10/11 = 240.

Say you’ve got a problem that, for a given integer n (0 < n ≤ 109), asks you to find the number of positive integers less than n and relatively prime to n. For example, for n = 12 we have 4 such numbers: 1, 5, 7 and 11.

The solution: The number of positive integers less than n and relatively prime to n equals to φ(n). In this problem, then, we need do nothing more than to evaluate Euler’s totient function.

Or consider a scenario where you are asked to calculate a function Answer(x, y), with x and y both integers in the range [1, n], 1 ≤ n ≤ 50000. If you know Answer(x, y), then you can easily derive Answer(kx, ky) for any integer k. In this situation you want to know how many values of Answer(x, y) you need to precalculate. The function Answer is not symmetric.

The solution here is to let res(i) be the minimum number of Answer(x, y) to precalculate, where x, y Î<1, …, i>. It is obvious that res(1) = 1, because if n = 1, it is enough to know Answer(1, 1). Let we know res(i). So for n = i + 1 we need to find Answer(1, i + 1), Answer(2, i + 1), … , Answer(i + 1, i + 1), Answer(i + 1, 1), Answer(i + 1, 2), … , Answer(i + 1, i).

The values Answer(j, i + 1) and Answer(i + 1, j), j Î<1, …, i + 1>, can be found from known values if GCD(j, i + 1) > 1, i.e. if the numbers j and i + 1 are not common primes. So we must know all the values Answer(j, i + 1) and Answer(i + 1, j) for which j and i + 1 are coprime. The number of such values equals to 2 * φ (i + 1), where φ is an Euler’s totient function. So we have a recursion to solve a problem:

res(1) = 1,
res(i + 1) = res(i) + 2 * j (i + 1), i > 1

Euler’s totient theorem:
If n is a positive integer and a is coprime to n, then a φ (n) º 1 (mod n).

Fermat’s little theorem:
If p is a prime number, then for any integer a that is coprime to n, we have

This theorem can also be stated as: If p is a prime number and a is coprime to p, then

Fermat’s little theorem is a special case of Euler’s totient theorem when n is prime.

The number of divisors:
If n = , then the number of its positive divisors equals to

For a proof, let A i be the set of divisors . Any divisor of number n can be represented as a product x1 * x2 * … * x k , where xi Î Ai. As |Ai| = ai + 1, we have

possibilities to get different products x1 * x2 * … * xk.

For example, to find the number of divisors for 36, we need to factorize it first: 36 = 2² * 3². Using the formula above, we’ll get the divisors amount for 36. It equals to (2 + 1) * (2 + 1) = 3 * 3 = 9. There are 9 divisors for 36: 1, 2, 3, 4, 6, 9, 12, 18 and 36.

Here’s another problem to think about: For a given positive integer n (0 < n < 231) we need to find the number of such m that 1 ≤ m ≤ n, GCD(m, n) ≠ 1 and GCD(m, n) ≠ m. For example, for n = 6 we have only one such number m = 4.

The solution is to subtract from n the amount of numbers, coprime with it (its amount equals to φ(n)) and the amount of its divisors. But the number 1 simultaneously is coprime with n and is a divisor of n. So to obtain the difference we must add 1. If n = is a factorization of n, the number n has (a1 + 1) * (a2 + 1) * … * (ak + 1) divisors. So the answer to the problem for a given n equals to

n – φ(n) – (a1 + 1) * (a2 + 1) * … * (ak + 1) + 1

Practice Room:
Want to put some of these theories into practice? Try out these problems, from the Topcoder Archive:

Baker, R., Weingartner, A.: Some applications of the double large sieve. Mon. Math. 170, 261–304 (2013)

Cai, Y.C.: On a Diophantine inequality involving prime numbers. Acta Math. Sin. 39, 733–742 (1996). (in Chinese)

Cai, Y.C.: On a Diophantine inequality involving prime numbers III. Acta Math. Sin. Engl. Ser. 15, 387–394 (1999)

Cai, Y.C.: A ternary Diophantine inequality involving primes. Int. J. Number Theory 14, 2257–2268 (2018)

Cao, X.D., Zhai, W.G.: A Diophantine inequality with prime numbers. Acta Math. Sin. 45, 361–370 (2002). (in Chinese)

Garaev, M.Z.: On the Waring–Goldbach problem with small non-integer exponent. Acta Arith. 108, 297–302 (2003)

Graham, S.W., Kolesnik, G.: Van der Corput’s Method of Exponential Sums. Cambrige University Press, Cambridge (1991)

Hua, L.K.: Some results in the additive prime number theory. Q. J. Math. 9, 68–80 (1938)

Heath-Brown, D.R.: The Pjateckiǐ-Šapiro prime number theorem. J. Number Theory 16, 242–266 (1983)

Ivic, A.: The Riemann Zeta-Function. Wiley, New York (1985)

Kumchev, A.: A Diophantine inequality involving prime powers. Acta Arith. 89, 311–330 (1999)

Kumchev, A., Nedeva, T.: On an equation with prime numbers. Acta Arith. 83, 117–126 (1998)

Piatetski-Shapiro, I.I.: On a variant of the Waring–Goldbach problem. Mat. Sbornik 30, 105–120 (1952). (in Russian)

Tolev, D.I.: On a Diophantine inequality involving prime numbers. Acta Arith. 61, 289–306 (1992)

Vinogradov, I.M.: Representation of an odd number as the sum of three primes. Dokl. Akad. Nauk SSSR 15, 61–64 (1937)

Zhai, W.G., Cao, X.D.: On a Diophantine inequality over primes. Adv. Math. 32, 63–73 (2003). (in chinese)

Zhai, W.G., Cao, X.D.: On a Diophantine inequality over primes(II). Mon. Math. 150, 173–179 (2007)

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Part of the Conjectures Are Not Theorems discussion is adapted from Mathematical Modeling: Teaching the Open-ended Application of Mathematics © Joshua Abrams 2000 and used with permission.

Translations of mathematical formulas for web display were created by tex4ht.