Prime numbers...

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09 Mar 2009, 6:58 pm

While I love math, I stopped when I reached calculus and have done little with it in many years. Not feeling well today, was letting my mind wander, and I had two questions that arose... is 1 considered to be a prime number, or an exception? I recall the in-class answer that a prime is a number whose only factors are one and itself - and that does not say that it cannot be one as well, but it is a curious situation. And are all prime numbers related by 2A + B, where A and B are both prime numbers? Perhaps there is a relationship between what A and B are, but for the first batch, assuming we know the first three (2,3,5):

2+2+3=7 (2A+B=D)
3+3+5=11 (2B+C=E)
5+5+3=13 (2C+B=F)
7+7+3=17 (2D+B=G)
7+7+5=19 (2D+C=H)
5+5+13=23 (2C+F=I)
13+13+3=29 (2F+B=J)
13+13+5=31 (2F+C=K)
17+17+3=37 (2G+B=L)
19+19+3=41 (2H+B=M)
19+19+5=43 (2H+C=N)
17+17+13=47 (2G+F=O)
23+23+7=53 (2I+D=P)
23+23+13=59 (2I+F=Q)
29+29+3=61 (2J+B=R)
31+31+5=67 (2K+C=S)
29+29+13=71 (2J+F=T)
31+31+11=73 (2K+E=U)
31+31+17=79 (2K+G=V)
23+23+37=83 (2I+L=W)
23+23+43=89 (2I+N=X)
43+43+11=97 (2N+E=Y)

Since I'm not versed in math, I'm sure this already has a name... but while it is not an absolute rule, doesn't this help refine the number set of those being examined as being prime numbers?

M.

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09 Mar 2009, 7:31 pm

makuranososhi wrote:

That is a rather interesting question. I will generalize it just a little bit. Does there exist an integer a > 1 such that S = {a*q1 + q2 | q1, q2 are odd primes}
contains all the primes except for a finite set of primes? I will research that a bit for you, but it sounds like it could be related to the Goldbach Conjecture or one of its variants.

The converse is not the case. For example 2*17 + 5 = 39 = 3*13 hence not a prime. But the original question where a = 2 sounds like a good problem.

Good eye!

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twoshots
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09 Mar 2009, 7:56 pm

And BTW, 1 isn't prime.

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makuranososhi
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09 Mar 2009, 8:02 pm

While I understand that it is an exception... can you explain better to be why it is excluded from the group? And I was not suggesting that all 2A+B scenarios would work, but that it appears that all primes (at least through 100) fall within a number set that is created in such a manner. Not all sums would be primes, but all primes would be a part of the number set created by the equation and therefore limiting the number of possible integers examined to find new primes.

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Orwell
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09 Mar 2009, 8:15 pm

Prime numbers are defined in such a way as to exclude 1. There isn't much reason for it; it's a definition.

Am I the only who dislikes prime numbers? They always seem so harsh and jagged.

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twoshots
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09 Mar 2009, 8:22 pm

makuranososhi wrote:

While I understand that it is an exception... can you explain better to be why it is excluded from the group?

Y'know, off the top my head I'm really not sure why. The most commonly cited reason on the internet is that it would complicate the fundamental theorem of arithmetic: each number would not have a unique prime factorization if 1 was allowed to be prime, e.g. 6=2*3=1*2*3. This site looks like it might have some interesting stuff to chew on

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makuranososhi
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09 Mar 2009, 8:24 pm

Appreciated! Going to keep going and see if I can brute force prove it further - I love math games, and this seems like an entertaining one for the moment.

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DNForrest
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09 Mar 2009, 8:30 pm

Orwell wrote:

Am I the only who dislikes prime numbers? They always seem so harsh and jagged.

As do I, with the exception of 2. I'm very OCD with 2's power series.

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09 Mar 2009, 8:34 pm

DNForrest wrote:

Orwell wrote:

Am I the only who dislikes prime numbers? They always seem so harsh and jagged.

As do I, with the exception of 2. I'm very OCD with 2's power series.

2's power series? What would that be....
2=2+x*0+x^2*0/2+x^3*0/6+...

Edit: wrong third derivative of 2 :doh:

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Last edited by twoshots on 09 Mar 2009, 8:39 pm, edited 1 time in total.

DNForrest
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09 Mar 2009, 8:38 pm

twoshots wrote:

DNForrest wrote:

Orwell wrote:

Am I the only who dislikes prime numbers? They always seem so harsh and jagged.

As do I, with the exception of 2. I'm very OCD with 2's power series.

2's power series? What would that be....
2=2+x*0+x^2*0/2+x^3/6+...

I think I may have used the wrong term, I meant:
2^x
x = 0, 1, 2, 3, 4, ...
So 1, 2, 4, 8, 16, ...

ruveyn
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09 Mar 2009, 9:31 pm

Orwell wrote:

Prime numbers are defined in such a way as to exclude 1. There isn't much reason for it; it's a definition.

Am I the only who dislikes prime numbers? They always seem so harsh and jagged.

All integers are composed (by multiplication) with primes ( and 1). That is why they are of interest.

Also prime numbers and factorization are the basis of public key - private key encription schemes.

Factoring is computationally hard compared to multiplication.

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makuranososhi
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09 Mar 2009, 10:44 pm

The remainder of the first hundred prime numbers, in the same format of 2A+B where A & B are both prime numbers as well.

47+47+7=101
43+43+17=103
47+47+13=107
53+53+3=109
53+53+7=113
61+61+5=127
59+59+13=131
59+59+19=137
61+61+17=139
71+71+7=149
71+71+9=151
73+73+11=157
73+73+17=163
47+47+73=167
47+47+79=173
83+83+13=179
89+89+3=181
89+89+13=191
73+73+47=193
89+89+19=197
97+97+5=199
97+97+17=211
103+103+17=223
107+107+13=227
109+109+11=229
113+113+7=233
113+113+13=239
113+113+17=241
107+107+37=251
107+107+43=257
113+113+37=263
113+113+43=269
127+127+17=271
127+127+13=277
127+127+17=281
127+127+19=283
127+127+29=293
151+151+5=307
149+149+13=311
151+151+11=313
149+149+19=317
151+151+29=331
157+157+23=337
167+167+13=347
173+173+3=349
173+173+7=353
173+173+13=359
181+181+5=367
181+181+11=373
181+181+17=379
173+173+37=383
173+173+43=389
197+197+3=397
197+197+7=401
199+199+11=409
191+191+37=419
199+199+23=421
197+197+37=431
197+197+39=433
211+211+17=439
211+211+23=443
223+223+3=449
227+227+3=457
227+227+7=461
229+229+5=463
227+227+13=467
233+233+13=479
233+233+17=487
239+239+13=491
241+241+17=499
233+233+37=503
233+233+43=509
251+251+19=521
241+241+41=523
269+269+3=541

M.

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09 Mar 2009, 11:11 pm

Orwell wrote:

Am I the only who dislikes prime numbers? They always seem so harsh and jagged.

Not so much dislike but rather uninteresting. The set is too dense to be fun. I like exponential series myself.

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Death_of_Pathos
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09 Mar 2009, 11:57 pm

Xelebes wrote:

Orwell wrote:

Am I the only who dislikes prime numbers? They always seem so harsh and jagged.

Not so much dislike but rather uninteresting. The set is too dense to be fun. I like exponential series myself.

Indeed. It was playing around with exponential series that I first noticed a relationship I now know to be defined by integration and differentiation.

What about Real Gaussian Primes?

makuranososhi, I suggest you learn Javascript. Enough can be learned in about an hour (Or less, with a tutor. In the absence of one I suggest "w3 schools") to allow you to programmatically attack this.

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10 Mar 2009, 12:00 am

What do you mean by exponential series? Power series?

And what's wrong with dense sets?

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10 Mar 2009, 12:37 am

Death_of_Pathos wrote:

Xelebes wrote:

Orwell wrote:

Am I the only who dislikes prime numbers? They always seem so harsh and jagged.

Not so much dislike but rather uninteresting. The set is too dense to be fun. I like exponential series myself.

Perhaps; it just caught my mind today and ran with it before dinner. It's an interesting tendency, but one that either I'll continue with or someone else may well in the future.

M.

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