A nifty theorem in geometry

[Bob Kolker 0]

Let ABC be a triangle with base BC and vertex A. Let the angle bisector of angle ABC intersect AC at point D. Let the angle bisector of angle ACB intersect AB at point E. If BD = CE then triangle ABC is isosceles.

I said the theorem was nifty (i.e. elegant). I didn't say it was easy to prove.

If the going is too rough look up Steiner-Lehmus Theorem for a proof. Interestingly enough this is a modern theorem (circa 1840). It was never in Euclid, although it seems to have same character as the famous pons asinorum (bridge of asses) theorem, Euclid I.5. What makes this theorem interesting is that there seems to be no direct synthetic proof. One must use reductio ad falsi (indirect proof) to get the result. If you find a direct synthetic proof then publish it and become famous overnight.

Bob Kolker

[Mindy Newton 0]

Let ABC be a triangle with base BC and vertex A. Let the angle bisector of angle ABC intersect AC at point D. Let the angle bisector of angle ACB intersect AB at point E. If BD = CE then triangle ABC is isosceles.

I said the theorem was nifty (i.e. elegant). I didn't say it was easy to prove.

If the going is too rough look up Steiner-Lehmus Theorem for a proof. Interestingly enough this is a modern theorem (circa 1840). It was never in Euclid, although it seems to have same character as the famous pons asinorum (bridge of asses) theorem, Euclid I.5. What makes this theorem interesting is that there seems to be no direct synthetic proof. One must use reductio ad falsi (indirect proof) to get the result. If you find a direct synthetic proof then publish it and become famous overnight.

Bob Kolker

Bob, does your interest in math extend to the theory prime numbers?

[Bob Kolker 0]

Bob, does your interest in math extend to the theory prime numbers?

It sure does. Every mathematician who has ever lived has a thing about prime numbers.

Bob Kolker

[Mindy Newton 0]

Bob, does your interest in math extend to the theory prime numbers?

It sure does. Every mathematician who has ever lived has a thing about prime numbers.

Bob Kolker

OK, try this out:

Every other number is odd, and every other other number is even. Both odd and even numbers arise at the same rate in the number series.

That rate, however, is, itself, either odd or even. Thus, it fits into one of those series and is alien to the other one.

As we know, the rate at which both odd and even numbers arise is two. You get one new odd number for each two you advance, from any starting point, and you get one even number for each two you advance.

Two is an even number, and it is the smallest even number. So, for the even number series, the smallest number is also the rate at which new members occur. Every new even number is some number of size-two steps beyond the first even number, and the even number series is--beyond its first member, universally divisible. It enjoys this symmetry because the rate at which even numbers arise is the same as the rate at which they multiply.

For the odd numbers, it is a different story. Odd numbers also arise at the rate of one for each two new numbers in the number series, but the smallest non-trivial odd number, 3, is larger than that rate of increase. As a result, odd numbers arise faster than they can multiply.[/i] It is this disjunction that is responsible for the existence and distribution of prime numbers!

Looked at from the other side, consider the smallest odd composite number. It is 9. Nine is the smallest odd number that can be expressed as the product of factors. (It might help some readers to recall here that only from multiplying odd numbers can one get an odd composite number.) Since 9 is the smallest composite odd number, any and all smaller odd numbers are not composites, that is, they have to be prime. The odd numbers smaller than 9 (leaving "one" out) are 3, 5, and 7; and 3, 5, and 7 are, in fact, prime numbers.

The next odd composite is 15--3 times 3 gives 9, and then 3 times 5 gives 15--which implies that the odd numbers smaller than 15, and larger than 9, are not composites, and must be prime. Those numbers are, of course, 11 and 13, and they are in fact prime. Continuing in this fashion, we note that 21 is the next odd composite number, and we check to see if the odd numbers that arise between 15 and 21 are prime, they are 17 and 19, and they are primes.

Twenty-five is the next odd composite, leaving only one prime to add to the list, 23. Twenty-five is the first odd composite that isn't factored by 3. It is, of course, the product of 5 and 5. Beginning with 25, the series of composite numbers becomes more dense. Up to 25, odd composites were simply the sequential products of 3 and the odd series of numbers, yielding 3 + 6n. Within the gap created by adding 6 to each composite, exactly two additional odd numbers arise.

Beginning with 25, a series defined by 5 + 10n of odd composite numbers is produced. As you've already begun to suspect, beginning with 7 times 7, 49, a third series of odd composite numbers, defined by 7 + 14n, is produced, and so on. The more general formula for each, prime-based series of odd composites will have occurred to you I'm sure: p + n2p.

As the series of odd composites becomes more dense, the complementary series, which is the series of primes, becomes less dense. In principle, however, the distribution of primes is completely predictable. Prime numbers are "created" by the discrepancy between the rate at which odd numbers arise and the rate at which they multiply.

That's about it. I haven't discussed this with anyone for years, so I might not have explained it well. You'll let me know if it doesn't seem to "add up?"

Mindy Newton

[Arnold 0]

That was well explained, but I have never understood the fascination of these relationships between numbers. Is there an end purpose, or is this just a game of sorts?

[TheOtherJake 0]

That was well explained, but I have never understood the fascination of these relationships between numbers. Is there an end purpose, or is this just a game of sorts?

Prime number research has applications in cryptography (code-making and code-breaking).

[Paul's Here 0]

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That's about it. I haven't discussed this with anyone for years, so I might not have explained it well. You'll let me know if it doesn't seem to "add up?"

Mindy Newton

That was quite fascinating and interesting. It's been decades since I studied math, but it was quite enjoyable to see the source of primes expressed that way. By the way, why is 1 not considered a prime? I always thought it was strange to omit it, since it is divisible by itself and 1. And if it were considered a prime number, how would it affect your explanation?

[Bob Kolker 0]

g and interesting. It's been decades since I studied math, but it was quite enjoyable to see the source of primes expressed that way. By the way, why is 1 not considered a prime? I always thought it was strange to omit it, since it is divisible by itself and 1. And if it were considered a prime number, how would it affect your explanation?

If 1 were admitted as a prime factorization would not be unique.

1 = 1*1 = 1*1*1 .. etc.

Bob Kolker

[Paul's Here 0]

g and interesting. It's been decades since I studied math, but it was quite enjoyable to see the source of primes expressed that way. By the way, why is 1 not considered a prime? I always thought it was strange to omit it, since it is divisible by itself and 1. And if it were considered a prime number, how would it affect your explanation?

If 1 were admitted as a prime factorization would not be unique.

1 = 1*1 = 1*1*1 .. etc.

Bob Kolker

Thanks. I've never seen that explanation before.

[Mindy Newton 0]

g and interesting. It's been decades since I studied math, but it was quite enjoyable to see the source of primes expressed that way. By the way, why is 1 not considered a prime? I always thought it was strange to omit it, since it is divisible by itself and 1. And if it were considered a prime number, how would it affect your explanation?

If 1 were admitted as a prime factorization would not be unique.

1 = 1*1 = 1*1*1 .. etc.

Bob Kolker

Thanks for the feedback and kind words.

If anybody is into divisibility, let me know. I have a scheme for "universal" divisibility that returns a factor. Note: it is not a factorization algorithm. It does not use any division itself, and looks, from the pseudocode, (I wrote the algorithm, and had someone write the pseudocode,) faster than division by a good bit, though not exponentially. I know very little about evaluating computability and such, I came across the divisibility algorithm from a number-theoretic point of view.

So if anybody knows about these things, let me know.

As far as why anybody would be interested in number patterns and such, it just is interestiing. I think you have to come onto a pattern yourself, and it grabs you. Trying to work out what a mathematician is pointing to in a math book is hard and frustrating, but if you get past the "wah?" factor, it is fascinating. I didn't study mathematics myself, but I did have excellent math teachers in high school. The principal happened to be a Ph.D. in math, and saw to it that we had excellent teachers and lots of courses. Two years of algebra, one of geometry and trig, and one of calculus. I exempted the Freshman math courses in college without even trying. A good high school is worth its weight in gold!

There are some fun pattern problems in a subject called odd-and-even permutations. If I can find the notes I have from a friend at Illinois State, who set the problems up for me, I'll put them up here.

Mindy