Maggie Johnson                                                                                                            Handout #24

CS103A

 

Proving Real Theorems

 

Key topics:

 

            * Why is Proof Important?

            * What Are We Trying to Prove?

            * The Art of Proving Things

            * A Review of Proof Strategies

* Direct Proof

            * Indirect Proof: Contradiction

            * Quantifiers I: Construction

            * Quantifiers II: Counterexample and Choose

            * Some Famous Conjectures

                                                                                                                                                           

 

 

“Mathematics, as a science, commenced when first someone, probably a Greek, proved propositions about “any” things or about “some” things without specification of definite particular things.”

                                                Alfred North Whitehead (1861-1947)

 

Consider the following questions:

 

1.      For any real number x, is floor(x-1) = floor(x) – 1?

2.      For any real numbers x and y, is floor(x-y) = floor(x) – floor(y)?

 

Think about these questions and see if you can convince yourself informally that your answer is correct.  We will return to them later.

 

We have just spent weeks learning about first-order logic and how to do both formal and informal proofs in FOL.  For the most part, the proofs we have done pertained to formal logic, e.g., we proved things about p, q, r, and tet(a).  We occasionally did a mathematical proof where we were interested in the actual meaning of the premises and the conclusion.  For example, we proved the sum of two even integers is another even integer. 

 

Now, we turn our attention to mathematical proofs.  Our goal is to use the foundations that we have laid in formal logic to help us derive solid proofs of meaningful statements.  We will begin by doing “semi-formal” proofs in this context where we provide detailed statement and reason charts. We call them semi-formal because they will not be as rigorous (and capable of mechanical checking) as the ones we did in Fitch.  But our semi-formal proofs will leave out no steps.  As we proceed, we will begin to work on the skills required to do informal proofs.  It takes practice learning what is appropriate to leave out in an informal mathematical proof.

 

Why is Proof Important?

 

Most students are introduced to the concept of proof in high-school geometry.  There, students learn that you have to do a formal proof about geometrical properties because:

 

            1) observation cannot prove because our eyes can deceive us:

                                                           

            2) measurement cannot prove because the certainty of the conclusion we arrive at is       dependent on the precision of the measuring instrument and care of the measurer.

 

            3) experiment cannot prove because the conclusions are only probable ones:

            It is probable that the dice are loaded if 10 successive 7's are thrown; it is even more     probable if 20 successive 7's are thrown (but it's not certain).

 

This last example is especially relevant in computer science.  Thus far, you have validated the correctness of the programs that you write by testing; but, in most cases, it is impossible to test all relevant inputs.  So, you do enough testing to convince yourself that the boundary cases and several cases in between are covered.  This is good enough for school assignments, but if you are designing the software for collision avoidance in the a new aircraft, you will want to do a lot more than just test isolated inputs; you will want to use proof techniques to verify correctness, as well as validate it.

 

In addition, proof is important to science and engineering students because it teaches us to think in very precise terms, which is a necessity in our field.  It also teaches a particular approach to  problem-solving and it strengthens algorithmic skills.  Donald Knuth in The Art of Computer Programming compared constructing a program from a set of specifications to writing a mathematical proof based on a set of definitions and known truths. 

 

 

What Are We Trying to Prove?

 

Statements that we have to prove often have the form of the conditional: if A, then B. To prove that such a statement is true, we have to analyze the meaning of the A statement and the B statement.  As shown in a previous lecture, the truth table for the conditional is:

 

                                                A         B          A->B

                                                T          T          T

                                                T          F          F

                                                F          T          T

                                                F          F          T

 

In propositional logic, we are not concerned with the meaning of A and B.  We analyze all the possible combinations of true and false (for A and B) to determine whether or not a compound proposition is true or false.  Now, however, we are concerned with meaning, so only one of the true/false combinations in the truth table will fit a particular A statement and B statement (with a resulting value for the conditional).  In a proof, we want to prove the truth of the conditional.  One way of doing this is to assume the premise is true, and we then proceed to show, using definitions and axioms from a particular frame of reference, that the conclusion is true.  If B is true, then the conditional is true (note that this method of proof "implies" causation of B by A, but as you will see, A does not have to be true). Or, we can just show A is false (then the conditional is always true), or we can show B is true without any information about A (if B is true, the conditional is always true).  There are lots of ways to do it, but the most common technique is using the A information to get to the B conclusion; when both are true, the conditional is true.

 

 

The Art of Proving Things

 

The reason we do proofs is to convince a skeptical audience that a given statement is absolutely true.  Imagine a listener challenging every statement that you make with "Why is that so?".  If you can counter with every possible challenge, your proof is valid.  As a very simple example, imagine having to prove to someone unfamiliar with math that if 5x+3 = 33 then x = 6.  Your proof might go like this:

 

            5x + 3 = 33                              given

           

            5x + 3 - 3 = 33 - 3                   we can subtract = quantities from both sides of an

                                                            equation

 

            5x = 30                                    subtraction

 

            x = 6                                        division

 

Of course, a proof for someone who is familiar with math would be much shorter since we could leave out the obvious steps (probably all of them).  An important point then, is know your audience.  What can you leave out based on their knowledge and experience?  The reasons that you feel must be included for the proof to flow, and to suit the needs of your audience come from a particular frame of reference.  The frame of reference of the above example is simple arithmetic.  Every frame of reference has basic definitions that you can accept as true and use in your proof.  Definitions are very important in proofs; they will frequently provide you with the starting point of a proof.  Most of the proofs we will do in the next couple classes will use number theory as the frame of reference.

 

As an example of how definitions can help in proofs: if you know that the definition of an even number n is n = 2k for some integer k, you can prove lots of things by substituting these definitions:

 

            Prove: If n is any even integer, then  (-1)n  = 1

 

            This statement is a conditional and as stated earlier, one way to proceed is to prove that             the conclusion is true.  To do this, we can use the information given in the premise.           Therefore, we know that n is an even number so we can substitute the definition of an           even number for n:

 

                        n is even: n = 2k for some integer k

 

                        (-1)n = (-1)2k                substitution

 

                        ((-1)2)k             property of exponents

 

                        (1)k                              (-1)2 = 1

 

                        (1)k = 1                        by the laws of exponents 

 

Knowing the definitions of the frame of reference, and being able to use them effectively is crucial in doing proofs:

 

            "Such then is the whole art of convincing.  It is contained in two principles: to

            define all notations used, and to prove everything by replacing mentally the

            defined terms with their definitions." 

Blaise Pascal

 

In the following sections, we will review the different methods of proof that we learned in FOL.  We will also provide examples of each in the number theory frame of reference.

 

 

 A Review of Proof Strategies

 

·        Convince yourself that the conclusion is indeed true by studying the premises and understanding their meaning:

 

 

 

 

Once you have convinced yourself that the conclusion is valid, there are two possible ways to proceed:

 

1.      If the process you went through in understanding the premises and conclusion (working through an example or defining an informal proof) gave you all the details you need, write up a formal (or informal depending on the requirements) proof following the same sequence of steps.  In writing an informal proof, you may leave out obvious steps if you know your audience can fill in those gaps.

 

2.      If you are convinced of the conclusion's validity but are having trouble formalizing it, try working backwards from the conclusion, rather than forward from the premises.  But always be careful that any intermediate goals you define are consequences of the available information.  Think carefully about each step, whether working forwards or backwards.

 

When finished, check your work:  Remember that each line in a proof (whether formal or informal) is either a premise, intermediate goal or final goal.  Any goal, intermediate or final, must be justified.  Make certain that the justifications are valid and lead step-by-step to the conclusion.

 

If you are writing a formal proof, the justifications must be explicitly stated.  If you are writing an informal proof, certain obvious justifications can be left out assuming that your audience can still follow the proof.

 

If any line in your proof is not a premise, intermediate goal or final goal (or perhaps a clarifying remark), it probably does not belong in your proof.

 

A well-written proof flows.  The reader should feel as though they are being taken on a ride that takes them directly and inevitably to the desired conclusion without any distractions or irrelevant details.

 

 

Direct Proof

 

Direct proof is a technique where we prove a conditional by showing that the conclusion B must be true when the premise A is true.  Therefore, the proof consists of attempting to find a link between the premises that we assume are true, and what we are trying to prove in the conclusion.  We called this technique “working backward”.

 

            Prove: If x and y are nonnegative real numbers that satisfy x+y = 0,

            then x = 0 and y = 0.

 

            Statement                                             Reason

            x >= 0                                                  given

 

            x = -y                                                   subtraction on given statement

 

            -y <= 0                                                            y >= 0 so therefore -y must be <= 0

 

            x <= 0                                                  substitution

 

            x = 0                                                    definition of a real number equaling 0:

                                                                        x <= 0 and x >= 0       

 

            0 + y = 0                                              substitution

 

            y = 0                                                    identity law

 

            Therefore, If x and y are nonnegative real numbers that satisfy x+y = 0,

            then x = 0 and y = 0.

 

Notice that the question that got us thinking backwards was: How can I show that a real number is 0?  We use a definition from the frame of reference (arithmetic) to answer this question: if x <= 0 and x >= 0 then x = 0. 

 

A real number is rational if and only if r = a/b for some integers a and b with b ? 0.  A real number that is not rational is irrational.

 

Prove: The sum of any two rational numbers is rational.

 

 

If n and d are integers and d ? 0, then n is divisible by d if and only if, n = d * k for some integer k.  The notation d | n is read “d divides n”.

 

Prove: For all integers a, b, and c, if a | b and b | c then a | c.

 

 

 

Indirect Proof:  Contradiction

 

The method of proof by contradiction is based on the fact that either a statement to be proven is true or it is false, but never both.  Suppose you can show that the assumption that a given statement is not true leads logically to a contradiction, impossibility or absurdity?  Then that assumption must be false; hence, the given statement is true. 

 

We see arguments by contradiction used in many real-life situations.  For example,

 

Suppose I did commit this heinous crime.  Then, at the time of the crime, I would have had to be at the scene of the crime.  In fact, I was getting married in front of 400 people at that moment as they will all testify.  This contradicts the assumption that I committed the crime.  Therefore, that assumption is false.

 

To do a proof by contradiction of a conditional, the first step is to assume the statement to be proven is false.  Then show that this assumption leads logically to a contradiction.  You may then conclude that the statement to be proved is true.

                                                      

Prove: If n is an integer and n2  is even, then n is even.

Assume: If n is an integer and n2  is even, then n is odd.

 

            Statement                                             Reason

            n = 2k + 1 for some integer k                definition of an odd number

 

            n2  = (2k + 1)2                          square both sides

 

            n2 = 4k2  + 4k + 1                                algebra

 

            n2  =  2 ( 2k2   + 2k) + 1                       algebra

 

            2k2  + 2k is an integer                           k is an integer

 

            n2 = 2k + 1                                           substitution

 

            n2   is odd                                             definition of odd number - contradiction

 

            Since we have arrived at a contradiction (n2 is odd when "n2 is even" is a given), we have

            proven: if n is an integer and n2  is even, then n is even.

 

Note the form of a semi-formal proof (whether direct or indirect) as illustrated above.  The statement to be proven is given (and in the case of proof by contradiction, the "assume" statement is also given); then a statement and reason chart is given with all detailed steps included.  Finally, a concluding statement is given summarizing why the statement is true.  Always include all three of these steps in your proofs on problem sets and exams  (if we request a semi-formal proof) to avoid losing points.

 

Prove or disprove: 1 + 3 v2 is irrational.

 

Another form of indirect proof is proof by contraposition.  Recall that the contrapositive of p ? q is ~q ? ~p.  So if we prove ~q ? ~p, it’s the same as proving the original statement.

 

An integer n>1 is prime if and only if, for all positive integers r and s, if n = r * s, then r = 1 or s = 1.  An integer n>1 is composite if and only if, n = r * s for some positive integers r and s with r ? 1 and s ? 1.

 

Prove: For all positive integers m and n, if m*n = 1 then m = 1 or n = 1.

 

Prove: For any integer a and any prime number p, if p | a, then p does not divide (a+1).

 

There are some classic little problems using contradiction.  Here is a famous one:

 

* Suppose there are only two types of people living on an island: knights and knaves.  Knights always tell the truth and knaves always lie:

 

            A says: B is a knight.

            B says: A and I are of opposite types.

 

What are A and B? To find out who's who, we have four possibilities: A & B are both knights; A is a knight and B is a knave; A is a knave and B is a knight; A and B are both knaves.   We'll start by assuming that A is a knight and see if we come up with a contradiction.

 

            If A is a knight, he always tells the truth so B is also a knight.  That means B also

            always tells the truth, but B says something contradictory.  So, A cannot be a knight.

 

            If A is a knave, then he is lying so B is also a knave.  B also lies  by saying they are

            of opposite types.  Therefore, A and B must both be knaves.

 

(From Smullyan: What is the Name of this Book?). By finding contradictions, we were able to find the solution.  This example is also important as an illustration of a proof by cases.  A proof by cases is where we must analyze more than one possibility to prove a particular statement.

 

 

Quantifiers 1: Construction

 

The next two topics concern conditionals that have quantifiers.  We will discuss existential conditionals first. They have the form:

 

            There is an object with a certain property such that something is true.    

 

The first step in proving these statements is recognizing the object, property, and the something that is true.  Then you proceed by using the construction method.  The basic idea is to just construct the object, which makes the statement true.  In addition, you must show that the object has the "certain property" and the "something is true".  But, we only need one object to prove the statement true.

 

Sometimes you construct the object by trial and error, or you might develop an algorithm to produce the desired object. 

 

            Prove: There exists an integer n that can be written in two ways as a sum of two prime   numbers.

 

            object: integer n

            property: none really except that n is an integer

            something is true: n can be written two ways as a sum of two prime numbers.

 

            By trial and error, n = 10:  5 + 5 = 10 and 7 + 3 = 10.  Therefore, there exists an integer n        that can be written in two ways as a sum of two prime numbers.

 

 

Quantifiers II: Counterexample and Choose

 

            For all x in S, if A(x) then B(x).

 

The first step in proving such a conditional is to rewrite the statement to be proven to get it into this form.  Then, we can proceed a couple different ways.  One is to find a counterexample thereby proving the statement is false (it doesn't work for all). 

 

            Prove: All primes are odd.

 

            New statement: For all integers k, if k is prime, then k is odd.

 

            2 is a prime number and is not odd.  Thus, by counterexample all primes are not odd.

 

 

Given any real number x, the floor of x , denoted floor(x), and the ceiling of x, denoted ceil(x), are defined as follows:

 

            Floor(x) = that unique integer n such that n <= x < n+1.

            Ceil(x) = that unique integer n+1 such that n < x <= n+1

 

 

Prove: For any real numbers x and y, floor(x-y) = floor(x) – floor(y).

 

Another method of proof is called the choose method, which is based on the following idea: To show that every element in a set satisfies a particular property, suppose an element x is a particular but arbitrarily chosen element of the set, and show that x satisfies that property.  The point of having x be arbitrarily chosen (or generic) is to make the proof general, i.e., you are making no special assumptions about x that are not also true of all the other elements of S.  You probably recognize this from FOL as universal introduction or generalization. 

 

So, whatever you prove about x, you can prove about any other element in the set.  Remember: You must be careful to keep the objects generic.

 

Prove: Any integer n > 1 is divisible by a prime number.

 

 

Some Famous Conjectures

 

There are many conjectures that have never been proven or disproven, some of which seem so simple that one might think they are easy to prove.  Some famous ones:

 

            1) Goldbach's Conjecture: If n is an even integer > 2, then n can be represented as the sum        of two primes.

 

            Note that 4 = 2+2, 6 = 3+3, 8 = 5+3, 10 = 5+5, 12 = 7+5, etc. No one has proven                 this but computer calculations show this is true for all integers up to 100,000,000.

 

            2) Euler's conjecture that a4  + b4  + c4  = d4 has no integer solution.  This was accepted as        true for 200 years until it was proven wrong by one counterexample:

 

                        958004 + 2175194 +4145604 = 4224814

                        (It was also proven wrong formally by Elkies at Harvard)

 

            3) Fermat's Last Theorem:  It is impossible to find positive integers x, y, z such that

            xn  + yn  = zn  for n >= 3 (there are solutions for n = 2 e.g., 32 + 42 = 52)

 

                        Fermat wrote this theorem in the margin of a book he was reading about 350

                        years ago.  He wrote beside it: "I have discovered a truly remarkable proof of

                        this theorem which this margin is too small to contain."  This was one of the

                        most famous open questions in mathematics until recently when it was proven by                                    Andrew Wiles of Princeton.

 

There will always be open conjectures in mathematics.  Recall that Gödel's Undecidability Theorem states that it is not always possible to find a proof for every true statement in a mathematical system.  Thus, we may never be able to establish whether certain conjectures are actually true.

 

 

More Proofs

 

The best way to sharpen your skills at doing proofs is to just try and do them.  The following problems can be used for additional practice.

           

1) You are visiting the island described above where only knaves and knights live.  You meet two natives Fred and Barney. Fred says: Barney is a knave. Barney says: Fred is a knave. Are either Fred or Barney a knave?

 

There is one knave.  Fred and Barney cannot both be knights because then both would be knaves (knights only tell the truth).  This is a contradiction.  Fred and Barney cannot both be knaves either because then both would be telling the truth which is impossible for knaves.  The only possible answer is that one is a knight and one is a knave - one is telling the truth and one is lying.

 

2) Give a direct proof of the following:  if a, b and c are integers and b is divisible by a and c is divisible by a, then (b-c) is divisible by a.

           

            Statement                                            Reason                        

            b = a * r for some integer r                definition of divisibility

            c = a * s for some integer s                definition of divisibility

 

            b - c = ar - as                                      substitution

 

            b - c = a(r - s)                                      distributive law

           

            r - s is an integer                                 r and s are both integers & so is  their difference

 

            b-c is divisible by a                              definition of divisibility

 

Therefore, if a, b and c are integers and b is divisible by a and c is divisible by a, then (b-c) is divisible by a.

 

3) Find the error in the following proof:  If a and b are even integers, then a * b is also an even integer.

                        Statement                                             Reason

                        a = 2 * c for some integer c                  definition of even

                        b = 2 * c for some integer c                  definition of even

 

                        a * b = 2c * 2c                                     substitution

 

                        2c * 2c = 2(2c2)                                   multiplication

 

                        2 * c * c is an integer                            product of integers is an integer

 

                        a* b = 2 * an integer                             substitution

 

                        a* b is even                                          definition of even

 

Using the same symbol c for the representation of a and b severely limits the generality of this proof.  By saying that a = 2c and b = 2c, the proof specifies that a = b and, therefore, we are only proving the statement for the case where a = b.  A correct  "choose method" proof must deduce the conclusion for all integers a and b, so we need a different symbol for b's representation.

 

4) Give a proof by contradiction of the following statement:  if a, b, & c are integers and if bc is not divisible by a, then b is not divisible by a.

 

            assume: if bc is not divisible by a, then b IS divisible by a.

 

            b = a * k for some integer k               definition of divisibility

 

            bc = (a * k)c                                       multiply both sides by c

 

            bc = a * (kc)                                       associative law

 

            kc is an integer                                                product of integers is an integer

 

            bc is divisible by a                               definition of divisibility (contradiction)

 

Since we have arrived at a contradiction (bc is divisible by a when "bc is not divisible by a" is a given), we have proven: if a, b, & c are integers and if bc is not divisible by a, then b is not divisible by a.

 

5) Give a proof by construction of the following:  There exists three consecutive odd prime integers a, b,  and c.

 

By trial and error: 3, 5 and 7.  Therefore, there exists three consecutive odd prime integers a, b,  and c.

 

 

 

Bibliography

 

G. Pólya, How to Solve It, Garden City, NY: Doubleday, 1957.

 

G. Pólya, Mathematical Discovery,  New York: Wiley, 1962.

 

R. Smullyan, What is the Name of This Book - The Riddle of Dracula and Other Logical Puzzles,    Englewood Cliffs, NJ: Prentice Hall, 1978.

 

D. Solow, How to Read and Do Proofs, New York: Wiley, 1982.

 

 

 

 

Historical Notes

 

If you are interested in how Wiles proved Fermat's Last Theorem, I have a brief description of the proof; send me some email or stop by during office hours.  The following is a report received just after the proof was announced:

 

Subject: Take That, Fermat!

 

Here is a recent report on Wiles' proof.

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News Item (June 23)-Mathematicians worldwide were excited and pleased today by the announcement that Princeton University professor Andrew Wiles had finally proved Fermat's Last Theorem, a 356-year-old problem said to be the most famous in the field.

 

Yes, admittedly, there was rioting and vandalism last week during the celebration. A few bookstores had windows smashed and shelves stripped, and vacant lots glowed with burning piles of old dissertations. But overall we can feel relief that it was nothing- nothing- compared to the outbreak of exuberant thuggery that occurred in 1984 after Louis deBranges finally proved the Bieberbach Conjecture.

 

"Math hooligans are the worst," said a Chicago Police Department spokesman.  "But the city learned from the Bieberbach riots. We were ready for them this time."

 

When word hit Wednesday that Fermat's Last Theorem had fallen, a massive show of force from law enforcement at universities all around the country headed off a repeat of the festive looting sprees that have become the traditional accompaniment to triumphant breakthroughs in higher mathematics.

 

Mounted police throughout Hyde Park kept crowds of delirious wizards at the University of Chicago from tipping cars over on the midway as they first did in 1976 when Wolfgang Haken and Kenneth Appel cracked the long-vexing Four-Color Problem. Incidents of textbook-throwing and

citizens being pulled from their cars and humiliated with difficult story problems last week were described by the university's math department chairman Bob Zimmer as "isolated."

 

Zimmer said, "Most of the celebrations were orderly and peaceful, But there will always be a few-usually graduate students-who use any excuse to cause trouble and steal. These are not true fans of Andrew Wiles."

 

Wiles himself pleaded for calm even as he offered up the long elusive proof that there is no solution to the equation xn + yn = zn when n is a whole number greater than two, as Pierre de Fermat first proposed in the 17th Century. "Party hard but party safe," he said, echoing the phrase he had repeated often in interviews with scholarly journals as he came closer and closer to completing his proof.

 

Some authorities tried to blame the disorder on the provocative taunting of Japanese mathematician Yoichi Miyaoka. Miyaoka thought he had proved Fermat's Last Theorem in 1988, but his claims did not bear up under scrutiny of professional referees, leading some to suspect that the fix was in. And ever since, as Wiles chipped away steadily at the Fermat problem, Miyaoka scoffed that there would be no reason to board up windows near universities any time soon; that God wanted Miyaoka to prove it.

 

In a peculiar sidelight, Miyaoka recently took the trouble to secure a U.S.  trademark on the equation "xn + yn = zn" as well as on the now-ubiquitous expression, "Take that, Fermat!" Ironically, in defeat, he stands to make a good deal of money on cap and T-shirt sales.

 

This was no walk-in-the-park proof for Wiles. He was dogged, in the early going, by sniping publicity that claimed he was seen puttering late one night doing set theory in a New Jersey library when he either should have been sleeping, critics said, or focusing on arithmetic algebraic geometry for the proving work ahead.

 

"Set theory is my hobby, it helps me relax," was his angry explanation. The next night, he channeled his fury and came up with five critical steps in his proof. Not a record, but close.

 

There was talk that he thought he could do it all by himself, especially when he candidly referred to University of California mathematician Kenneth Ribet as part of his "supporting cast," when

most people in the field knew that without Ribet's 1986 proof definitively linking the Taniyama Conjecture to Fermat's Last Theorem, Wiles would be just another frustrated guy in a tweed jacket teaching calculus to freshmen.

 

His travails made the ultimate victory that much more explosive for math buffs.  When the news arrived, many were already wired from caffeine consumed at daily colloquial teas, and they took to the streets en masse shouting, "Obvious!  Yessss! It was obvious!"

 

The law cannot hope to stop such enthusiasm, only to control it. Still, one has to wonder what the connection is between wanton pillaging and a mathematical proof, no matter how long-awaited and

subtle.

 

The Victory Over Fermat rally, held on a cloudless day in front of a crowd of 30,000 (police estimate: 150,000) was pleasantly peaceful. Signs unfurled in the audience proclaimed Wiles the greatest mathematician of all time, though partisans of Euclid, Descartes, Newton and C.F. Gauss and others argued the point vehemently.

 

A warmup act, The Supertheorists, delighted the crowd with a ragged song, "It Was Never Less Than Probable, My Friend," which included such gloating, barbed verses as- "I had my proof all ready/But then I did a choke-a/Made liberal assumptions/Hi! I'm Yoichi Miyaoka."

 

In the speeches from the stage, there was talk of a dynasty, specifically that next year Wiles will crack the great unproved Riemann Hypothesis ("Rie-peat!  Rie-peat!" the crowd cried), and after

that the Prime-Pair Problem, the Goldbach Conjecture ("Minimum Goldbach," said one T-shirt) and so on.

 

They couldn't just let him enjoy his proof. Not even for one day. Math people.  Go figure 'em.

 

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Note: if you are interested in other famous unsolved problems, check out:

 

R.K. Guy, Unsolved Problems in Number Theory, New York: Springer-Verlag, 1980.