Except for 2 and 3, all prime numbers are one more or one less than some multiple of 6:
A…B…C…D…E…F
0…1…2…3…4…5
6…7…8…9…10…11
12…13…14…15…16…17
18…19…20…21…22…23
24…25…26…27…28…29
30…31…32…33…34…35
36…37…38…39…40…41
42…43…44…45…46…47
48…49…50…51…52…53
54…55…56…57…58…59
etc.,etc.,etc.
A=6x
B=6x+1
C=2x
D=3x
E=2x
F=6x-1
What to contributions to science and mathematics have been anticipated, to the chagrin of fellow dopers?
All prime numbers?
Can you show that this is true for all prime numbers?
Yes, easily. Anything else would be either divisible by 2 or 3. As the OP demonstrates at the bottom of his post.
Similarly, with the exception of 2 and 5, all prime numbers have a last digit of 1, 3, 7, or 9. With the exception of 3 and 5, all prime numbers are 1, 2, 4, or 8 above or below a multiple of 15. Etc.
To the OP: I know your thread isn’t actually meant to be about this result in particular, but your last sentence got a bit garbled and I’m not sure what it is meant to be about, so… 
Trivially. 6x, 6x + 2, 6x + 3 and 6x + 4 are all divisible by 2 or 3.
Actually, on re-read, the sentence is pretty clear. I’m guessing the thread is supposed to be about results dopers have discovered excitedly and then, to their chagrin, realized were already well known or trivial?
In my childhood at one point, I got excited about the fact that 1 + 2 + 3 + … + n = n (n+1)/2, which I proved by thinking about the fact that it would have to average out to halfway between the top and the bottom number added. Very mundane now to any math-oriented person’s eyes, I imagine, but at the time, I was all worked up for a week or two over finding similar results.
I independently rediscovered the Newson-Raphes (no idea how to spell that) method for square root approximation. Basically, take your best (or really, any reasonable) guess for the square root of x, and average that with x/that guess.
Newton-Raphson (yes, that Newton) and it works for the roots of any (differentiable) function.
In naive theory. You can construct pathological counterexamples, or even just fuck yourself by picking too far off a guess to start with. There’s a whole body of results on when it will and won’t work out. But certainly, the algorithm can be applied to any differential function; whether it actually approximates an answer or keeps shifting around is the only question.
When I was in about fourth grade, I proved to myself using diagrams that x^2 - 1 = (x-1)(x+1). I was so proud of myself!
Ed
Do you mean you did this by considering the areas of rectangles of appropriate side lengths? If so, Euclid would be proud of you. That’s how the ancient greeks seem to have done their algebra (ie. geometrically), and book II of his Elements is full of that kind of stuff.
I recall being thrilled to discover that if you wrote down the sequence of squares and took their differences, you got the sequence of odd numbers. Another thing the predecessors of the Sumerians probably figured out.
I discovered a way of tessellating pentagons when I was in high school. A few years later I sent it off to the wolfram math website to ask if it was original. About a year later they got back saying it was a subtype of one of the known types.
You have a pentagon ABCDE
Sides AB, BC, CD, and DE are all the same length
The angles of B and D are 90 degrees
The angle of C is more than 90 and less than 180
An example is shown here, the middle one three down from the top.
In college, I was rather proud (and a little surprised) to have taught the school how not to convert from decimal to hexidecimal by way of conversion to binary first. I’ve no idea why they did it that way – it was totally pointless using binary as a middle-man. So I gave them the equation:
Decimal to Hex for 8-bit value:
Left digit: x=y/16
Right digit: x=16*(MOD(y/16))
So, a binary value of 228: Left digit is 228/16 or 14, expressed in hex as $D. Right digit is (MOD(y/16)) which is 0.25, multiplied by 16, which is 4. So 228 in hex is $D4.
Multiple byte values can be done in stages using the same idea, and conversion the other way is even easier:
y=left digit*16+right digit
This all came as news to the professors there, and frankly, I think that says more about the profs than any level of ingenuity I might have displayed. It wasn’t all that clever to me, it was just the way it seemed most logical to be done.
Just a bit of pedantry, but what you give isn’t a conversion from decimal to hexadecimal; it’s a conversion from numbers qua numbers to their hexadecimal representation. You could combine it with the rules for treating decimal representations as numbers which we’re all so familiar with, but that’s not intrinsic to the method you give. Indeed, just swapping out the 16s for any other base, you’ve given the general way to convert from any base to any other base (well, as long as the result is two digits long).
(Similarly, 228 isn’t a binary value; it’s just a number (and “228” is a decimal representation of that number)).
1 + 3 + 5 + . . . + 2k+1 = (k+1)^2
I conjectured this and then proved it one night while lying in bed. The proof was by induction.
Upon waking and taking up pen and paper of course, it turned out to be trivial to do as a simple algebraic manipulation.
The question arose when I was reading 2001: a Space Odyssey or a sequel. Bowman was observing the relative dimensions of the monolith. The ratio was 1:4:9 in the three spacial dimensions we can see but he was marveling at how the pattern continued into higher dimensions.
Since it wasn’t spelled out, I had to wonder if he was referring to the sequence of squares or some other simple sequence that had the same first three terms. It seemed to me that it could be the sum of the first N odd integers. It was then necessary for me to prove that the two series where in fact identical before I could go to sleep.
Doh! Sorry, I meant “a decimal value of 228.” I had binary on the brain I guess.
But yes, actually, you’re right in that you could swap in any base system I suppose. I just never worked with anything other than decimal, binary and hex, so I didn’t consider anything else.
Like 74westy, I “discovered” as a youth the fact that adding up odd numbers creates the sequence of square numbers.
It puzzled me that where x^2=y, (x+1)(x-1)!=y. So I worked on the general case and worked out that (x+1)(x-1)=(x^2)-1; and furthermore that (x+z)(x-z)=(x^2)-(z^2). When I reduced the formula it came down to x=x. So much for my career in mathematical innovation.
Are you deflated to find that your discovered reduction ended up reaching something as trivial as x = x? Don’t be discouraged by that; naturally, the simplified end of any proof will always be something immediately true. The innovation is in seeing the reduction itself.
I had discovered that, too, by noticing a pattern in examples:
7^2=49…68=48
10^2=100…911=99
(X^2-1)=(X+1)(X-1)
In a much earlier post I theorized that, in trigonometry, any triangle with all integer value (not fractions or decimals) side lengths would not have angles with all integer values; conversley, a triangle with all integer value angles would not have all integer value side legths.
I rediscovered that “difference of two squares” thing once while I was out riding my bicycle. It was only when I was explaining it to my Dad later that he pointed out that I’d derived x[sup]2[/sup] - y[sup]2[/sup] = (x + y)(x - y), a piece of math I already knew perfectly well. :smack:
