baric

The fossil record shows multiple tool-making hominid species living concurrently as our ancestors were on their way to world dominance. We wiped them out through competition. If given time, chimps would develop increasing dexterity as their tool use became more complex. The fact that they are already so close is a strong indication of how unexceptional it is. They are intelligent, self-aware, and can manipulate primitive tools with their hands. If tool use provides a selective advantage, the species will become increasingly proficient with it. Intelligence evolves where it provides a selective advantage, not as it is needed. A modern jelly fish does not have a brain, but some of their ancestors evolved them. Technological intelligence can be very common yet technological ETs remain very rare. It's the last variable in the Drake Equation.

John Barrow is wrong. The evidence on Earth suggests that primitive intelligence is inevitable once you have evolved larger life forms as we have observed a wide variety of species that have complex social interactions and language. It is tool use that seems to spark the rise of technological intelligence. That level of intelligence, however, is a lot like the movie 'Highlander' -- there can be only one. This is because technological species expand so rapidly that they effectively stunt the advancement of others. So the real question is "how common is tool use?". There are several examples of this among Earth species, so I suspect it is also very common in the universe. If humans magically disappeared tomorrow, we'd likely have technological chimpanzees in another two million years. Pierre Boulle was right.

Not really. The space environment is out there, but it's a stretch to say that we live in it. The density of energy sources available to us drops off dramatically as soon as you leave the Earth's surface. For example, there are no fossil fuels outside of Earth. There are no hydroelectric sources of power. There is no surface to establish wind power that does not melt the underlying structure. There are no ready sources of uranium to refine for nuclear power. We still have solar power, of course. There would need to be a huge technological jump to find new sources of energy that would provide even slow growth beyond the confines of Earth, much less exponential growth.

Yeah. I was looking at the book's result that you quoted and assumed they were using the formula for a finite series. That's what their result looks like, anyway.

ok, I get a different answer from the book, but trying to sum based on the geometric series you provided. However, I come up with a different answer from yours as well. We are doing something wrong! if given: [math]A_{r} = a^{n+1} + (ar)^{n+1} + (ar^{2})^{n+1} + \cdots + (ar^{x-1})^{n+1}[/math] then [math]A_{r}(r^{n+1}) = (ar)^{n+1} + (ar^{2})^{n+1} + \cdots + (ar^{x})^{n+1}[/math] thus [math]A_{r} - A_{r}(r^{n+1}) = a^{n+1} - (ar^{x})^{n+1}[/math] [math]A_{r} (1 - r^{n+1}) = a^{n+1} (1 - (r^{x})^{n+1})[/math] [math]A_{r}= a^{n+1} \frac{ (1 - (r^{x})^{n+1})}{1 - r^{n+1}}[/math]

Well, that was never my intent and I apologize for any offense. I can only judge someone's proficiency in a subject by the content of their posts. If I were asking physics questions, I certainly hope that someone would talk down to my more basic level of understanding and would not consider it condescending. The OP seemed stuck on the supposed significance of summing digits and even referenced Vedic mathematics, of all things, so I was actually trying to lightly chide our illustrious Dr. Rocket for perhaps posting over the OP's head.

/hug I tried several times to explain very simply why your "law" was just an artifact of our base system. Dr Rocket presented a proof demonstrating the same thing. Was his proof sufficient, or do you still feel that this discovery is a "Law of Primes"?

If that was directed at me, I would like to point out that proficiency in mathematics specifically requires the "privilege of education". There have been many courteous attempts to explain to the OP why this discovery is not as significant as he would like to believe -- to correct his "misdirection", as it were. Tailoring your answer to the education level of the questioner is not patronizing. It's teaching.

I agree completely! Unfortunately, the intended audience for this proof would not understand why you chose 10, that funny triple-equals symbol, or a 1. They might not even understood QED!

That's a great analogy, and one describing what I also feel is the most likely solution the paradox. Also, everyone gets only three balls to toss.

Moore's Law is not a law. It's an observation about the exponential rate of growth in the early stages of a new technology. We don't live in an environment that can sustain unlimited exponential growth.

Stop right there. Uri Geller is a certified fraud. Here's a classic takedown by James Randi....

oh, that makes sense Thanks!

No, it is NOT a law of prime numbers. It is an artifact of using base-10 to represent numbers. If you used base-5, for example, your "law" would no longer work. And since base-10 is just a human preference for displaying numbers, it cannot be a requirement for any mathematical law. I am not familiar with this technique. Can you demonstrate how it works on the following multiples of 7? 14, 91, 133

In addition, all of these numbers are very rough hypothetical estimates based on mathematical models. They are as accurate as my attempts at throwing darts... in the right direction, but that's about it. Arguing over precision is extremely premature.

One trillion degrees for a neutron star is already a hypothetical estimate, so remember that you are dealing with large uncertainties at these extremes. The bottom line is that not all questions have answers, especially those dealing with extreme conditions (temperature, velocity, gravity, pressure, etc) that are nearly impossible or extraordinarily expensive to reproduce.

I would have had to go to wikipedia for those!

Yes, plasma is another state of matter. There are a lot more than four, though. That's what I was trying to get across Let's not forget Einstein-Bose Condensates!

Solid, liquid, vapor, supercritical fluid and superfluid are all observed states of matter.

But it's not really a "law" of primes as it is dependent upon representing the numbers in base-10. It doesn't work in base-5, for example.

Just so you are aware, these digit summation rules will all help you rule out certain numbers for primality. They work because we use base-10 to represent numbers: Because 2 & 5 are factors of 10: Because 9 & 11 are adjacent to 10: I hope that helps. You can write numbers out in different bases and verify that these same rules apply.

Given those previous two statements, don't you think that's a really big IF? (it's not correct) All measurements of the speed of light have been constant. The notion of a speed of light "decay" is an old creationist attempt to explain away the age of the universe as given by redshift measurements of distant galaxies

That is an interesting article! Specifically, they found dense disk containing a lot of water vapor, which stars are forming from. But because they are observing through the radio spectrum, they can't really determine what percentage of the disk is water vapor. Given the distribution of elements, the disk will still be dominated by Hydrogen and Helium. But knowing that these clouds contain water vapor (at least near the galactic centers) gives us another way to find these star-forming clouds.

There is definitely a reason why the composites seem to be distributed the same as primes in the original sieve. In particular, what you described as If you examine your algorithm, you will see the list of composites in Prime2 are all numbers divisible by 2. This is obvious, of course. All composites in your Prime3 as divisible by 3, but NOT by 2. All composites in your Prime5 list are divisible by 5, but not by 2 or 3. Do you see what you are doing? You are simply filtering the set of integers for each subsequent iteration of the prime sieve, which artificially creates a sparser distribution for each subsequent prime sieve. However, this uneven distribution does NOT match the distribution of primes in the original sieve. Whereas the original sieve results in an increasingly sparse distribution of primes, your sub-sieves necessarily have a repeating distribution because they are based on the fixed number of preceding primes. For example, your Prime5 sieve shows composites divisible by 5 that are not divisible by 2 and 3. This means that these composites will simply start at 5*5 (25) and cycle every 30 integers (2*3*5). Thus: 25, 35, 55, 65, 85, 95, 115, 125, 145, etc. In other words, this distribution will not decrease in frequency like the prime sieve so therefore it is not the same.

I understand that English is probably not your first language, but the problems in your grammar make it difficult to parse the philosophical meaning that you are trying to get across in your post. Science is the only non-dogmatic process man has developed to increase his knowledge about the world. Ultimately, no scientific theory is sacred and each one risks being overturned or refined as more evidence is gathered. In other words, science and reasoning are the hallmarks of freethinking. There are no "gods", "prophets", or "idols" in science.