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 between addition and multiplication martin Junior Fellow  Posts: 14 Threads: 1 Joined: Jul 2008 03/12/2011, 09:22 AM (03/11/2011, 10:59 PM)bo198214 Wrote: Hey Martin, we are not talking about the function f(x) = a  x but about the function g(x) = a [x] b. Right, I didn't remember correctly - but it had to do with considering only an interval for which arithmetic progression turns into geometric progression and then extrapolating to obtain further values. I've worked on so many different ideas during the last few years, so I didn't dig into it any further. (The most proliferate idea being a real number Y for which floor(Y*p#) always produces primes for p=prime>1 with a probability of at least 1-10^-560) nuninho1980 Fellow   Posts: 95 Threads: 6 Joined: Apr 2009 03/14/2011, 05:04 PM (03/10/2011, 09:10 PM)lloyd Wrote: This has probably been thought of before, but here goes anyway. I was thinking about the "sesqui" operation intermediate between adding and multiplying; I'll write "@" here. Obviously a @ b should lie between a+b and ab. Maybe we should take the mean. But which one, arithmetic or geometric? Since one applies to addition and the other to multiplication, why not take both? Then we'll take the mean of these two. But which mean? Again, take both; the proposed value for the sesqui-operation is then the limit of this process when iterated many times. In fact the two values converge quite quickly and for 10-digit precision we usually have convergence within 3 or 4 iterations. Here are some values for a @ a: (...) I discovered this forum after asking a question recently on sci.math. It looks like people here have been thinking about the same thing: I asked if the next operation after exponentiation should require new numbers, the way that addition/subtraction, multiplication/division, exponentiation/root-taking/logarithms lead from the counting numbers to negative, real and complex numbers respectively.I'm very interesting about operation "@". but what's your code (pari/gp, maple...)? lloyd Junior Fellow  Posts: 10 Threads: 1 Joined: Mar 2011 03/14/2011, 05:33 PM So I tried this out on the weekend to see what it looked like, and it works very well. Between addition and multiplication I used the "weighted arithmetic-geometric mean" that I described earlier in the post. Between multiplication and exponentiation I used a similar weighted function I call the "weighted exponential-geometric mean". Here is the C code; I lazily had it iterate 6 times instead of calling itself recursively and ending when the two means were close enough. I calculate m = a * b and n = a ^ b, I specify phi with some value between 1 and 2 and I call this function: float wgxm(float m, float n, float phi) { int i; float p,q; phi--; for (i=1;i<6;i++) { p = pow(n/m,phi) * m; q = pow(m, pow(log(n)/log(m),phi)); m = p; n = q; } return p; } The attached graph shows the functions y = x {phi} 3, where phi varies from 0 to 2 by stepping 0.2. The light blue lines are at integer values of phi, i.e. y=x+3, y=3x and y=x^3. The yellow lines are the values calculated by the function above (and the corresponding one between addition and multiplication. The x axis goes from 1 to 20 and the y axis goes from 0 to 800. The grid lines are every unit in the x direction, and every 40 units in the y direction. Something needs to be done to clean up the exponential/geometric interpolation function a little for it to work with values close to x=1 (the log of values close to 0 is negative). I don't think the formula works quite right whenever x^3 is smaller than x*3. The second attached graph is a close-up of the first, where x goes from 1 to 2 and y goes from 1 to 9, with gridlines in both directions every 0.25 (meaning the stretching/scaling is only by a factor of 2). I'll have to see how the wgxm function looks if I interpolate between 3x and 3^x next, instead of x^3. Let me know if you think any of this looks nice. (03/11/2011, 06:12 PM)JmsNxn Wrote: (03/11/2011, 12:35 AM)lloyd Wrote: Surely, though, {0.25} should be weighted 3/4s towards the arithmetic mean, and 1/4 towards the geometric mean. Ah but is the weighting carried out arithmetically or geometrically? Apply a 3/4 arithmetic : 1/4 geometric weighting there too! And take the limiting case again. In other words, for a {0.25} b, with a

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