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Mathematicians Team Up To Close the Prime Gap
Hugh Pickens DOT Com writes "On May 13, an obscure mathematician garnered worldwide attention and accolades from the mathematics community for settling a long-standing open question about prime numbers. Yitang Zhang showed that even though primes get increasingly rare as you go further out along the number line, you will never stop finding pairs of primes separated by at most 70 million. His finding was the first time anyone had managed to put a finite bound on the gaps between prime numbers, representing a major leap toward proving the centuries-old twin primes conjecture, which posits that there are infinitely many pairs of primes separated by only two (such as 11 and 13). Now Erica Klarreich reports at Quanta Magazine that other mathematicians quickly realized that it should be possible to push this separation bound quite a bit lower. By the end of May, mathematicians had uncovered simple tweaks to Zhang's argument that brought the bound below 60 million. Then Terence Tao, a winner of the Fields Medal, mathematics' highest honor, created a 'Polymath project,' an open, online collaboration to improve the bound that attracted dozens of participants. By July 27, the team had succeeded in reducing the proven bound on prime gaps from 70 million to 4,680. Now James Maynard has upped the ante by presenting an independent proof that pushes the gap down to 600. A new Polymath project is in the planning stages, to try to combine the collaboration's techniques with Maynard's approach to push this bound even lower. Zhang's work and, to a lesser degree, Maynard's fits the archetype of the solitary mathematical genius, working for years in the proverbial garret until he is ready to dazzle the world with a great discovery. The Polymath project couldn't be more different — fast and furious, massively collaborative, fueled by the instant gratification of setting a new world record. 'It's important to have people who are willing to work in isolation and buck the conventional wisdom,' says Tao. Polymath, by contrast, is 'entirely groupthink.' Not every math problem would lend itself to such collaboration, but this one did."
We cannot allow a prime gap!
sometimes its better to go it alone, then come back to the group with your results so that someone else may profit from them.
sometimes its better to be a part a group in order to establish your ideas and discuss, then go it alone when the group holds you back.
If they keep this shit up, pretty soon they will prove that every number is prime.
'It's important to have people who are willing to work in isolation and buck the conventional wisdom,' says Tao. Polymath, by contrast, is 'entirely groupthink.' Not every math problem would lend itself to such collaboration, but this one did."
History is rife with examples of the lone genius making a leap forward, thereby allowing the crowd to take it even further. See The Structure of Scientific Revolutions by Thomas Kuhn.
>> which posits that there are infinitely many pairs of primes separated by only two (such as 11 and 13)
Yawn. Call me when you find a set of primes separated by one.
Will they ever learn to factor prime numbers though? I understand it's difficult, but solving it would save a lot of embarrassment when people misstate the problem.
Er...2 and 3. What do I win?
Now James Maynard has upped the ante by presenting an independent proof that pushes the gap down to 600. A new Polymath project is in the planning stages, (...) to push the bound even lower.
600 ought to be enough for anyone.
Set your phasers on "funky"!
Three people are asked to prove that all of the odd numbers are prime - a physicist, a mathematician and a programmer.
The physicist goes first. "3 is a prime, 5 is a prime, 7 is a prime, 9 is a ... oops, experimental error, 11 is a prime ...".
Next the mathematician takes a crack at it: "3 is a prime, 5 is a prime, 7 is a prime, and the rest by induction".
Finally it's the programmer's turn. "3 is a prime, 5 is a prime, 7 is a prime, 9 is a prime, 11 is a prime ...".
Was it just me or did anyone else have a hard time following that summary? At first I thought it was Yitang Zhang who settled "a long-standing open question". But the first sentence is actually talking about the eight - James Maynard.
So in summary, if a pair of primes is defined by one following the other, it was theorized that we would find an infinite number of such pairs separated by 2. Various people have proven that gap to be from 70m, 60m, 4680, and now 600. Thank you James Maynard.
They are not so sexy, after all...
You will find 2 prime numbers within 600 of another prime pair.
That is, basically, the theory, yes. But if we can get that number down to "2" it proves a centuries-old conjecture that could lead to all sorts of interesting proofs of other things becoming true also.
In terms of computers:
You do realise that we use the difficulty of, in particular, finding large prime numbers as the basis for most modern security protocols implemented on computers? Precisely BECAUSE it's so hard to do?
We're not talking about 2, 3, 5, 7, etc. but we're talking about primes with MILLIONS of digits. Primes so large that even to prove they are prime can take a long time. Primes so enormous that multiplying two of them together makes a number that's almost impossible to factorise back to the correct original primes, so much so that we use it as the basis for things like SSL, TLS, etc.?
And, no, computers can't do mathematical proof. They can help as tools but they are dumb. You do not prove that every number to infinity has a prime within, say, 600 numbers by printing out every number. By definition you'll be there until infinity on even the fastest possible machine in the universe. You could prove that primes up to a number X that would hold true, but X would never be sufficient to prove it was always true. Just the fact that primes only have to be N numbers apart before you hit the next one could lead to mathematics that might well accelerate the discovery and manipulation of primes themselves.
But if you come up with a clever mathematical proof that GUARANTEES the answer, no matter what X is or how many billions of digits it has, without having to worry about ever finding a *particular* prime, then you have something that a mathematician can take as "fact" and incorporate into larger proofs about the universe. Imagine if we just assumed that every prime was like this, and applied it to a large scale engineering project, and then found out that actually - once you get past a few billion atoms - the premise doesn't hold? It'd be catastrophic.
The last "proof by computer" (i.e. by brute-force rather than as a tool) was the four-colour theorem. And even that was just because the problem could be reduced (by a mathematician, and using other proven theories, and logical inference) to a few thousand cases that the computer could iterate. It was used as a time-saver back in the days of manual calculation, not mathematical proof.
If N is between 2 and 10^260, then the number of primes less than N is more than N / 600. So in that range the _average_ gap between consecutive primes is less than 600. For N = 10^20 it is actually quite rare that the gap between two consecutive primes is over 600.
There's none. the number of primes smaller than n is équivalent to n/ln(n) when n goes to infinity (thanks to Hadamard and Vallee Poussin theorem). If there was a upper bound for two successive primes, it wouldn't be the case.
Does anyone happen to know what the greatest known lower bound is? (i.e. the largest known difference of two successive primes?)
There is none.
Proof: Select an arbitrarily large number N. The numbers between (N! + 2) and (N! + N) are all composite ((N! +2) is divisible by 2, (N! + 3) is divisible by 3, ..., and (N! + N) is divisible by N). Since you can find an arbitrarily large span of composite numbers, there is no upper bound on the gaps between primes.
QED.
No, they are saying that you can always find a pair of primes separated by 600. Let's say you list all the primers between 2 and N. You enumerate all the pairs whose difference is 600. What they are saying is that if you look beyond N, you will always find another such pair. They are NOT saying how much further you have to look.
They are *not* saying that given any prime number p, then p+600 is also prime.
Their goal is to demonstrate that the same is true for 2 instead of 600.
No we don't.
Primality testing is easy - the problem is in P. Approximate methods for finding primes are very efficient. Exact checking is rarely used.
Modern security protocols rely on the problem of factoring a number into primes being difficult. Or on inverting exponentiation within a prime field.
Slashdot: where don knuth is an idiot because he cant grasp the awesome power of php
It looks like you don't understand what GP was asking (at best) or you don't understand the summary/primes.
I think the GP was asking if there are always less than 600 between primes. The answer to his question is "no". The higher you go the larger gaps can be between primes. There can be untold millions/billions/trillions etc. between two distinct primes. This proof shows not that there are never more than 600 between primes, but that there are an infinite number of pairs of primes that are separated by less than 600. The difference is small but important. There may be two primes separated by a vast number, yet the higher you go there will always be a pair of primes coming up that are separated by less than 600.
For example:
The numbers
2^57,885,161 - 1
and
2^43,112,609 - 1
are primes. They have 17,425,170 and 12,978,189 digits in them. They are the largest two primes we know of. They are separated by a bunch of numbers in between them, almost 5,000,000 DIGITS (note digits not numbers) and all the numbers between them are composites. HOWEVER, the next largest prime may simply be (2^57,885,161 1) + 600 because there will always be a chance that there is a prime coming up less than 600 away from the current highest prime.
This is getting closer and closer to proving the long held belief/hope that there are an infinite number of primes separated by only 2. NOT that EVERY prime is separated by 2 from every other prime. That would be obviously false. Simply that there are an infinite number of primes salted throughout all those impossibly high ones that are only 2 apart.
No, the maximum distance grows without bounds. What this proves is that you can always find two more primes that are less than 600 apart, so the minimum distance does not grow without bounds. It has absolutely nothing to do with the distance between one pair of primes and another pair.
Live today, because you never know what tomorrow brings
And, no, computers can't do mathematical proof. They can help as tools but they are dumb. You do not prove that every number to infinity has a prime within, say, 600 numbers by printing out every number. By definition you'll be there until infinity on even the fastest possible machine in the universe. You could prove that primes up to a number X that would hold true, but X would never be sufficient to prove it was always true. Just the fact that primes only have to be N numbers apart before you hit the next one could lead to mathematics that might well accelerate the discovery and manipulation of primes themselves.
But if you come up with a clever mathematical proof that GUARANTEES the answer, no matter what X is or how many billions of digits it has, without having to worry about ever finding a *particular* prime, then you have something that a mathematician can take as "fact" and incorporate into larger proofs about the universe. Imagine if we just assumed that every prime was like this, and applied it to a large scale engineering project, and then found out that actually - once you get past a few billion atoms - the premise doesn't hold? It'd be catastrophic.
What you say has nothing to do with computers. Why would anyone program a computer to work case-by-case like that? It's just as futile as going case-by-case by hand. Likewise, if one is inclined to generate higher-level, logical proofs by hand, then why not program a computer to generate higher-level, logical proofs? Oh wait, that's been done for decades (eg. AUTOMATH, or the entire field of Automated Theorem Proving).
The last "proof by computer" (i.e. by brute-force rather than as a tool) was the four-colour theorem. And even that was just because the problem could be reduced (by a mathematician, and using other proven theories, and logical inference) to a few thousand cases that the computer could iterate. It was used as a time-saver back in the days of manual calculation, not mathematical proof.
Erm, what lets you define "proof by computer" as "by brute-force"? Are you claiming that all computer programs are brute-force? That's clearly nonsense. Are you claiming that a computer running an efficient algorithm is just a 'tool' and that the Mathematical ability actually exists in the algorithm's programmer? If so, you must also claim that Deep Blue's programmers are much better chess players than Deep Blue. In that case, why weren't they the world champions?
Also, the brute-force 'proof' of the Four Color Theorem, from 1976, was unsatisfactory to many people. It only proved the Four Color Theorem under the assumption that the program is correct, but nobody could verify such an assumption. In 2005 a new proof-by-program was constructed, but this time the program was written and verified in Coq. Only a tiny bit of code needs to be verfied in order to trust this proof (Coq's implementation of the Calculus of Inductive Constructions), and since this code is shared by all Coq users it's already had many eyes on it since appearing in the mid 1980s.
No, that's not the theory at all....
The theory is that no matter how high you look, you can always find 2 prime numbers within 600 of each other.
i.e. For any number X, there exists a pair of prime numbers Y, Z where Z>X and Y>X and Z-Y600
It's entirely possible that having found Y,Z, there are no other primes anywhere near those two.
You do realise that we use the difficulty of, in particular, finding large prime numbers as the basis for most modern security protocols implemented on computers? Precisely BECAUSE it's so hard to do?
We're not talking about 2, 3, 5, 7, etc. but we're talking about primes with MILLIONS of digits. Primes so large that even to prove they are prime can take a long time.
That's not how public-key crypto works at all.
Generating large primes is dead easy. This is exactly what Java keytool and OpenSSL to every time you ask them to generate a private key: they generate two large prime numbers (hundreds of digits each, not millions) and the private key is those two numbers, while the public key is the product of those two numbers. What makes this kind of crypto hard to crack is the difficulty of factoring the product of two large primes: the search space is absolutely humongous. Determining whether a given number is prime or not is much easier.
Their goal is to demonstrate that the same is true for 2 instead of 600.
The _real_ hypothesis is this: Given _any_ pattern, like (p, p+2, p+6, p+8) where it isn't obvious that only a finite number of solutions exist, there will be an infinite number of primes following that pattern.
A case where there is obviously a finite number of solutions is (p, p+4, p+8) because one of the three numbers must be divisible by 3. Or any pattern involving an odd number like (p, p + 1027); either p or p + 1027 must be even so except for p = 2 they can't be both primes.
No, the maximum distance grows without bounds. What this proves is that you can always find two more primes that are less than 600 apart, so the minimum distance does not grow without bounds. It has absolutely nothing to do with the distance between one pair of primes and another pair.
A simple proof: If you take a large number n, then n! + 2 is divisible by 2, n! + 3 is divisible by 3, and so on until n! + n which is divisible by n. n! + 1 and n! + n + 1 might be primes, but none of the numbers in between. So we have a gap between prime numbers of at least n.
I call BS. That gap is only N-2.
so little of what news is dragged before me these days does much to make me hopeful of humanity's prospects on this planet. This story is the rare exception. We could be a great species. We could solve what looked for centuries to be impossible problems. We could...
/. This story was not in any of my regular channels today.
Thanks
SLASHDOT: news for people who can't concentrate on work or have no life at all and got tired of yelling back at the TV.
It's people like you who make me want to learn more about maths. I felt like there could be an arbitrarily large gap between primes, but I had no idea how to express it. What you wrote, in addition that there are infinitely many primes proves the point perfectly.
Is 1563649 a prime number?
No, that's not the theory at all. The theory does not say there is always a prime within 600 of another (that's simply not true).
The theory says for any number X, there is a pair of primes larger than X within 600 of each other. That pair may be 2 larger than X, 12 larger than X, or 21,515,359 larger than X.
Everything else you said is pretty much spot on though.
The induction step and base case are obvious. The proof as laid out is correct for an arbitrary N. The induction step is to show that it is also true for N+1. Then your base case is to show that it is true for a specific N and N+1 like N=3 and N=4 (trivial to verify). At that point it is proven for all N where N is in the set of Natural Numbers and N >=3.
Honestly I thought it was a very well formed comment for Slashdot. You shouldn't flame something just because you don't understand it. And if you are looking for strong rigor, this is Slashdot, not a mathematical journal. Anybody with 2+ years of undergrad college math should have been able to complete the proof without even hitting up a Google search.
Only N-2... for arbitrary N. Pick your claimed bound as (natural number) x. I will choose N = x + 4 (I'll not only hit your bound, but do better). Then there is a gap of length x+2, which is bigger than what you claimed is possible.
Enjoy the troll feed.
or have quite a number of difficult and important theorems been proven in the last couple decades? Fermat's last theorem, the Poincare conjecture, now lots of progress on this conjecture? What have mathematicians been doing right recently?
No.
What this theory says is that no matter how far up you look on the number scale, you can always find a pair of larger primes that are separated by less than 600.
i.e. for any number X you always find primes larger than X that are closer than 600 from each other
In the opposite direction (what is the maximum gap between primes), the gap increases without bound.
i.e
For any number X you can always find closest primes that are more than X apart.
Here's a proof:
Take any number N
N! = (N) x (N-1) x (N-2) x...x (3) x (2) x(1) (i.e. N times itself minus 1 times itself minus 2, etc....the factorial of N)
N! is not prime...it is divisible by all numbers from 1 to N by definition.
N!+2 is not prime...it's divisible by 2 (remember N is divisible by 2 and 2 is divisible by 2)
N!+3 is not prime...it's divisible by 3
.
.
.
N!+N is not prime...it's divisible by N.
That means none of the numbers between N!+2 and N!+N are prime, so we have a gap of at least N-2.
This is true for ANY number N, so we can always find a gap as large as we want.
This is not a proof by induction it is a proof by contradiction, no induction step is needed.
It assumes there is a number N such that their must be at 2 primes between M and M + N, for any M, then the proof goes on to show how to pick a M for which this is not the case.
unless you are referring to the proof that the numbers between N! and N! + N are divisible not primes (clearly they are since you can write it as a*k+k = a*(N + 1) where a*k=N! for all values of k between 1 and N ). But you don't need induction to prove that either.
It isn't trying to cover every prime, just saying that no matter how far you go along them, you'll always be able to find another "nearby pair" further on.
The holy grail of the exercise is to bring the 600 down to 2, so that we'll be able to say there are an infinite number of so-called twin primes.
Say d(n) = p(n+1) - p(n).
Write down d(n) as a list of numbers.
It'll bounce around all the time (see other posts for proof that you can always find a value as large as you like), but you'll also always be able to find a value that's 600 or less, and at that point you've found a "nearby pair."
The holy grail of the full twin prime conjecture is just saying that there are an infinite number of 2s in the sequence of d(n)s.
You've got to find them, though...
Contradiction works too. I just always found induction to be more intuitive personally. Honestly any proof technique that can be applied to a given problem (and it is often the case that more than one technique is easily used to prove a give problem) still results in a mathematically valid proof. I reworded it into a proof by induction because that makes sense to me and I was just filling in some minor blanks. A proof by contradiction would work too, but it just isn't my style.
all the numbers between them are composites.
Ahem. Those are the two largest known primes (because primes of that form are particularly easy to search for using existing techniques), but there's nothing to say that there are not unknown primes between them. In fact, there almost certainly are many; the density of primes in that region should be on the order of 1 in every 100 million integers, so there are probably at least about 10^17425161 other primes in that span.
Does anyone happen to know what the greatest known lower bound is? (i.e. the largest known difference of two successive primes?)
There is none.
Proof: Select an arbitrarily large number N. The numbers between (N! + 2) and (N! + N) are all composite ((N! +2) is divisible by 2, (N! + 3) is divisible by 3, ..., and (N! + N) is divisible by N). Since you can find an arbitrarily large span of composite numbers, there is no upper bound on the gaps between primes.
QED.
Wrong set. You're dealing with ALL primes. The question is about the set of KNOWN primes (you know, the ones listed in the NSA's Big Book of Primes). Between the known primes, there is a greatest known lower bound.
When our name is on the back of your car, we're behind you all the way!
Theoretically, it can't be any lower than 2. The fascinating thing is that as prime numbers become larger, they are found further and further apart, which plotted as a graph is more like a log n curve. But every now and again, you find a couple that are just two units apart. Usually one of them is something like (2^n)-1 and the other is (2^n)+1 . If the first one is written out as binary, it would form a prime number of 1's eg. 31. The only way such a binary number could have factors is one with 2^(n-1) number and the other being an odd number with 1 spaces a prime number distance apart.
Simplest example would be 3 and 5. 3 = (2^2)-1, and 5 = (2^2)+1
Vintage computer adverts: http://www.vintageadbrowser.com/computers-and-software-ads
I understood some of those words.
It gripped her hand gently. 'Regret is for humans,' it said.
The linked abstracts are pretty vague. Are there any mathematicians here who can explain how (seemingly arbitrary) large numbers like 600 or 70 million come out of these proofs? People are saying they're all tweaks of the same basic method, so what is that basic method, exactly?
Visit the
Perhaps, he was educated as to the stupidity of his remark later.
You are being MICROattacked, from various angles, in a SOFT manner.
Note if you can find the proof of this, then you have killed multiple birds with one stone.
You get the infiinite twins problem solved.
You get the Goldback conjecture solved.
And you find that is also shows that there are infinite sets
of twin primes separated by any distance.
Which means there are infinite quad primes as I mentioned above.
You are being MICROattacked, from various angles, in a SOFT manner.
For example:
The numbers
2^57,885,161 - 1
and
2^43,112,609 - 1
are primes. They have 17,425,170 and 12,978,189 digits in them. They are the largest two primes we know of. They are separated by a bunch of numbers in between them, almost 5,000,000 DIGITS (note digits not numbers) and all the numbers between them are composites.
Wrong.
There are for sure a lot of prime numbers between 2^43,112,609 - 1 and 2^57,885,161 - 1. We don't know about any specific, but there exists theorem stating that there is always prime between x and 2x, where x is positive integer. So there is always a prime between 2^i and 2^(i+1). So there are at least
57885161-43112609=14772552 (+-1) primes in this range.
...does this affect encryption in some way? My understanding is that a lot of encryption relies on the difficulty of finding prime numbers. I may be wrong. (It's certainly not my specialty.)
Explanation by car analogy:
You're driving on a highway leaving a city. At every prime numbered mile marker there's a gas station. As you leave the city the gas stations are close together, with a station at the 2 mile marker, another at the 3 mile marker, another at the 5 mile marker, etc. As you get into the suburbs the gas stations are less frequent. As you get into the desert you find that gas stations are hard to find.
But you notice something - it seems that no matter how far you drive into the desert, you occassionally find gas stations just two miles away from each other. You start calling these pairs "twins". Now someone has told you there are an infinite number of gas stations on the road, but you're wondering if there are an infinite number of twins. Will there always be more twins in front of you, or at some point will you have past the last pair? The Twin Primes Conjecture suggests there will always be another pair of twin gas stations on the road, but it has never been proven.
Well, you think, so that's never been proven, but I've also noticed that sometimes I see gas stations just 4 miles apart. Does anyone know whether that will stop occurring? At some point will I pass the last pair of gas stations that are 4 miles apart? Again the answer is, nobody knows. How about 6? 8?
Until recently, the answer was always, nobody knows. Then this Chinese guy proved that if you're looking for pairs of gas stations that are less than 70 million miles apart, there will always be such a pair on the road in front of you.
Then somebody else proved that there would also always be a pair of gas stations in front of you less than 5000 miles apart. Most recently, someone proved that you would always have in front of you some pairs of gas stations that are within 600 miles of each other.
We still suspect that will can always look forward to finding more twin gas stations, pairs that are within 2 miles of each other, but we don't have a proof yet.
I often don't like the choices people make, but I like the fact that people make choices. That's why I'm a conservative.
Suppose that the set of primes P is dense in R. Then for all x,y in R there exists a p in P such that x p y. Let x=0 and y=1, observe that there is no prime number greater then zero and less than one, contradiction. Therefore P is not dense in R.
This sig is not paradoxical or ironic.
Why isn't cold fjord in here blaming Snowden for all this?
In fact, thanks to Bertrand's Postulate, we can be certain there are millions of primes between those two.
What an amazing discovery! Is there a pattern to the distribution to such pairs of primes? Are they composed of compound sets or is their distribution entirely unpredictable? When people make such discoveries or proofs, we get the chance in our lifetime to pose the next round of questions, the ones that might take decades (or centuries) to answer. Great news item!
And, no, computers can't do mathematical proof. They can help as tools but they are dumb. You do not prove that every number to infinity has a prime within, say, 600 numbers by printing out every number. By definition you'll be there until infinity on even the fastest possible machine in the universe. You could prove that primes up to a number X that would hold true, but X would never be sufficient to prove it was always true. Just the fact that primes only have to be N numbers apart before you hit the next one could lead to mathematics that might well accelerate the discovery and manipulation of primes themselves.
But if you come up with a clever mathematical proof that GUARANTEES the answer, no matter what X is or how many billions of digits it has, without having to worry about ever finding a *particular* prime, then you have something that a mathematician can take as "fact" and incorporate into larger proofs about the universe. Imagine if we just assumed that every prime was like this, and applied it to a large scale engineering project, and then found out that actually - once you get past a few billion atoms - the premise doesn't hold? It'd be catastrophic.
What you say has nothing to do with computers. Why would anyone program a computer to work case-by-case like that? It's just as futile as going case-by-case by hand. Likewise, if one is inclined to generate higher-level, logical proofs by hand, then why not program a computer to generate higher-level, logical proofs? Oh wait, that's been done for decades (eg. AUTOMATH, or the entire field of Automated Theorem Proving).
The last "proof by computer" (i.e. by brute-force rather than as a tool) was the four-colour theorem. And even that was just because the problem could be reduced (by a mathematician, and using other proven theories, and logical inference) to a few thousand cases that the computer could iterate. It was used as a time-saver back in the days of manual calculation, not mathematical proof.
Erm, what lets you define "proof by computer" as "by brute-force"? Are you claiming that all computer programs are brute-force? That's clearly nonsense. Are you claiming that a computer running an efficient algorithm is just a 'tool' and that the Mathematical ability actually exists in the algorithm's programmer? If so, you must also claim that Deep Blue's programmers are much better chess players than Deep Blue. In that case, why weren't they the world champions?
Also, the brute-force 'proof' of the Four Color Theorem, from 1976, was unsatisfactory to many people. It only proved the Four Color Theorem under the assumption that the program is correct, but nobody could verify such an assumption. In 2005 a new proof-by-program was constructed, but this time the program was written and verified in Coq. Only a tiny bit of code needs to be verfied in order to trust this proof (Coq's implementation of the Calculus of Inductive Constructions), and since this code is shared by all Coq users it's already had many eyes on it since appearing in the mid 1980s.
Very true in the sense that a programmer (and mathematician) that might not be good at solving certain complex problems that requires lots of short term memory with frequent storage and retrieval all within a finite time, can figure out a way to solve the problem of solving that problem.
The proof as laid out is correct for an arbitrary N. The induction step is to show that it is also true for N+1
Appart for the major woosh, as you didn't get the obvious joke "this holds for any N" -> "no !!! only for N-2" (and I'm not sure at this point that you will even get the hint) :)
You seem to have a major problem understanding the induction process which you claim to be your prefered and most intuitive way of understanding mathematical proofs. (but since you're a nice person you still admit that your GP's post is good enough for slashdot standards (thank you very much for him/her and the rest of us))
So "The proof as laid out is correct for an arbitrary N" as you said... Why in the world would you need to show that it is also true for N+1 ? N+1 *is* an arbitrary N
Induction is about having specific working examples (not arbitrary !!!) and proving that from here the next candidates must also be valid.
Also, induction is not just about having N and proving N+1, or I could 'prove' you many funny things (sometimes you need more than just one element to get the next and depend on several ones (P(0) and P(1) are true, thus P(2) (depending on the previous two) is true)