We didn’t know randomness could be “subtle”

_{Denyse O'Leary

March 22, 2016

Mathematics, News}

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Erica Klarreich at Quanta has the story of a surprising new result about prime numbers from Kannan Soundararajan and Rober Lemke Oliver. They have found that, given a prime number with a certain last digit, there are different probability for the last digit of the next one (among the various possibilities). This violates usual assumptions that such things are in some sense “random”, indicating just how subtle this “randomness” is. More.

From Klarreich at Quanta:

Two mathematicians have uncovered a simple, previously unnoticed property of prime numbers — those numbers that are divisible only by 1 and themselves. Prime numbers, it seems, have decided preferences about the final digits of the primes that immediately follow them.

Among the first billion prime numbers, for instance, a prime ending in 9 is almost 65 percent more likely to be followed by a prime ending in 1 than another prime ending in 9. In a paper posted online today, Kannan Soundararajan and Robert Lemke Oliver of Stanford University present both numerical and theoretical evidence that prime numbers repel other would-be primes that end in the same digit, and have varied predilections for being followed by primes ending in the other possible final digits.

…

The discovery is the exact opposite of what most mathematicians would have predicted, said Ken Ono, a number theorist at Emory University in Atlanta. …

See also: Infinity at Starbucks: Starring Laszlo Bencze and Art Battson

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Comments

Mapou: "All Darwinists (the so-called evolutionary biologists) should be thrown in jail for perpetrating such a monumental hoax on the world and for indoctrinating our children with their blatant lies." I hope that you are not being serious. The idea of throwing people in jail for their ideas is communist and anti-Christian. If you are serious about this then you are in some pretty evil company.Indiana Effigy_{March 23, 2016
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Mung:
Mapou:
The only problem with this is that the combinatorial explosion kills any stochastic search mechanism dead. This includes RM+NS. Why? Because the search space increases exponentially with the number of variables.
Every time you say this I see fingers going into ears and tongues wagging LALALALALALALA.
Yeah. It completely refutes the theory at its source. Any other theory would have never seen the light of day in the face of such outrageous stupidity. And yet, Darwinism persists. Why? Because it is a religion, that's why. All Darwinists (the so-called evolutionary biologists) should be thrown in jail for perpetrating such a monumental hoax on the world and for indoctrinating our children with their blatant lies.Mapou_{March 23, 2016
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IE - See the below article, you might find it of interest: http://www.evolutionnews.org/2014/10/the_dog_delusio090751.htmlspecter13_{March 23, 2016
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Kairosfocus, I understand that selection is an elimination process; one of culling, as you call it. But haven't humans produced chihuahuas and bull mastiffs using the same process in a relatively short period of time? If our only record of these two breeds was a fossil record, we would certainly identify them as being related, but do you really think that we would classify them as the same species? And chihuahuas breeding with each other will continue to produce variations. But they are unlikely to give birth to a mastiffIndiana Effigy_{March 23, 2016
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Mapou:
The only problem with this is that the combinatorial explosion kills any stochastic search mechanism dead. This includes RM+NS. Why? Because the search space increases exponentially with the number of variables.
Every time you say this I see fingers going into ears and tongues wagging LALALALALALALA.Mung_{March 23, 2016
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IE, I am focussed on where NS is held to be a factor of necessity, i.e. if not fit enough, a variety dies out in the competition. There is of course always a stochastic element in something like that, discussed in Darwin but practically the result is culling out. Further to that, the process is a culler out of varieties and so by definition is a remover of information expressed in the inferior varieties that die out. It is not a creative element, a source of bio information. That is still left to the engines of variation, up to 47 by one count. KFkairosfocus_{March 23, 2016
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I'm sorry, but I am still confused. Are you suggesting that natural selection is random? It would seem to me that any selection process (other than random selection) would be non random. I understand that random processes, on their own, cannot produce complexity. But it seems to me that any non random process has the possibility of creating complexity.Indiana Effigy_{March 23, 2016
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IE, forgive, EZ is trying to suggest that my use of blind watchmaker chance and/or necessity as summarising the driving forces evolutionary materialism has to appeal to to explain OOL and origin of body plans across the tree of life somehow leaves out "natural selection" which is often said to be non random and it is held to transform the power of chance variation. It does not and it highlights that the forces are blind, unintelligent and confronted by a massive blind search for islands of function in vast configuration spaces. What explains say finch beak variations and circumpolar gulls or the like faces a very different order of challenge accounting for origin of birds. Where of course natural selection means that in competitive environments varieties that are sufficiently inferior get culled out, die off as they do not reproduce successfully enough to keep around. KFkairosfocus_{March 23, 2016
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KairosFocus: "EZ, so if culling — subtraction of variation — in part driven by differential reproductive success is non random, does it not then come under not logical..." I am embarrassed to admit that I don't understand your point. Could you please make it more clear?Indiana Effigy_{March 23, 2016
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You assume to know what the initial replicator was like.
Show me the initial replicator. Or else shut up.mike1962_{March 23, 2016
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EZ, so if culling -- subtraction of variation -- in part driven by differential reproductive success is non random, does it not then come under not logical [If A then B must hold . . . ] but instead mechanical, blind . . . as in, non foresighted . . . necessity? (I leave off the random, stochastic aspects of such differential success.) KFkairosfocus_{March 23, 2016
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ellazimm:
You make a lot of assumptions which may or may not be true. You assume to know what the initial replicator was like. AND, more importantly, you assume that there had to be a random search algorithm to progress from the simplest replicator to the more complex ones. And if you’re wrong . . . I think you cannot be sure of those assumptions.
I don't have to be sure of any assumption since I'm not making any that I'm aware of. The truth is that a random searcher does not make any assumption either. It does not know how big the search space should be for anything. This makes it even worse because the search space might as well be infinite. New genes have to come from somewhere and if you don't know how many base pairs you need or even why you need them, the problem is much worse. But even if one assumes a starting point using the most basic replicator, it could not be much simpler than a bacterium. Decreasing the number of initial base pairs to 10,000 would not make a difference.
You seem to be arguing against a concept that no evolutionary theorist holds. That forms arose through a random search process.
So evolutionary theorists know (by some magic revelation that they the rest of us idiots are not privy to, I guess) that the initial search was not random? Where is the science in that? Where is the falsifiability? Who or what was directing the search, pray tell? And why should anybody believe evolutionary theorists? What makes them so special? I personally think they are idiots.
Conclusion: Darwinism is a cretinous religious cult for people who don’t understand simple math.
I understand the simple math and I disagree with you. And I don’t have to be rude and disrespectful of your view while I do that.
I make it a point to always be rude, contemptuous and disrespectful to all dirt worshippers because this is the way they have treated those who have disagreed with them in the past. This is the way they always treat their critics. In their view, if you are not an atheist, a Darwinist or a materialist, you are an idiot or a stupid creationist. We saw an example of that during the recent debate with that jackass/crackpot/ignoramus, Lawrence Krauss. You people need to get a taste of your own medicine. And you will get it. You are stupid as dirt.Mapou_{March 23, 2016
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ellazimm: I had experiences immediately before and after the period when I was sedated. But not during. How would you express that?
"I don't remember my experiences, if any, during the interval."
ellazimm: How is is that a physical stimulation (being administered a sedative) induces a period of amnesia? More than once. And it’s not just me. Most people under general anaesthesia have no memories of the time their are ‘under’.
And many do; see #28. About "no memories": future science will figure it out — does that sound familiar? :). Anyway, the main point is that we have to consider several alternative explanations to the one you offer.Origenes_{March 23, 2016
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ba77 #24
I don’t argue with non-persons, since all rationality is lost when a person denies he is real.
You are allowed to be prejudiced. But it doesn't make you right. KF #25
Ez, and which part of chance variation plus natural selection escapes blind chance and mechanical (material, non logical, blind force) necessity? KF
That would be selection which un-randomly singles out forms which are better able to cope with the natural environment. Origenes #27
“I have no experience” constitutes a contradiction. To be able to observe that one has no experience presupposes experience.
I had experiences immediately before and after the period when I was sedated. But not during. How would you express that?
From “no memories” one can hypothesize that a period without experience has passed. However there are other explanations; amnesia being one of them. There are other (better) explanations which I won’t address at this point.
How is is that a physical stimulation (being administered a sedative) induces a period of amnesia? More than once. And it's not just me. Most people under general anaesthesia have no memories of the time their are 'under'. Mapoo #29
The only problem with this is that the combinatorial explosion kills any stochastic search mechanism dead.
That's assuming an initial, highly complex start to life. What if the initial replicator was very simple?
This includes RM+NS. Why? Because the search space increases exponentially with the number of variables. The simplest bacterium has at least 100,000 base pairs.
Who said the initial replicator was a bacterium?
This means that the mutation search space is 2^100,000. You could have a computer the size of trillions of universes running for trillions of years at trillions of cycles per second and it would not make a dent in that search space. Now imagine the size of the search space for a complex creature like a frog or a human being.
You make a lot of assumptions which may or may not be true. You assume to know what the initial replicator was like. AND, more importantly, you assume that there had to be a random search algorithm to progress from the simplest replicator to the more complex ones. And if you're wrong . . . I think you cannot be sure of those assumptions. You seem to be arguing against a concept that no evolutionary theorist holds. That forms arose through a random search process.
Conclusion: Darwinism is a cretinous religious cult for people who don’t understand simple math.
I understand the simple math and I disagree with you. And I don't have to be rude and disrespectful of your view while I do that.ellazimm_{March 23, 2016
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ellazimm:
You forgot to mention millions of years of non-random selection which favoured ‘systems’ which were better able to exploit their surroundings and conditions.
The only problem with this is that the combinatorial explosion kills any stochastic search mechanism dead. This includes RM+NS. Why? Because the search space increases exponentially with the number of variables. The simplest bacterium has at least 100,000 base pairs. This means that the mutation search space is 2^100,000. You could have a computer the size of trillions of universes running for trillions of years at trillions of cycles per second and it would not make a dent in that search space. Now imagine the size of the search space for a complex creature like a frog or a human being. Conclusion: Darwinism is a cretinous religious cult for people who don't understand simple math.Mapou_{March 23, 2016
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F/N: http://www.reuters.com/article/us-dreaming-anesthesia-idUSCOL06020420070220kairosfocus_{March 23, 2016
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ellazimm: When I’m under anaesthetic I have no experiences, no memories.
"I have no experience" constitutes a contradiction. To be able to observe that one has no experience presupposes experience. From "no memories" one can hypothesize that a period without experience has passed. However there are other explanations; amnesia being one of them. There are other (better) explanations which I won't address at this point.Origenes_{March 23, 2016
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BA77, Welcome back. "I don’t argue with non-persons, since all rationality is lost when a person denies he is real." I don't think that you can conclude that if the mind is purely material that means that you are a non-person.Indiana Effigy_{March 23, 2016
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Ez, and which part of chance variation plus natural selection escapes blind chance and mechanical (material, non logical, blind force) necessity? KFkairosfocus_{March 23, 2016
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I don't argue with non-persons, since all rationality is lost when a person denies he is real.bornagain77_{March 23, 2016
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KF #20
am I speaking to an OS cobbled together by blind chance and mechanical necessity? If so, on what grounds do I have any right to expect response to reason or decision and ability to freely follow a reasoned argument and conclude per ground consequent rather than mechanically necessary cause effect with maybe some chance randomness tossed in?
You forgot to mention millions of years of non-random selection which favoured 'systems' which were better able to exploit their surroundings and conditions. In that context being analytic and able to evaluate complicated situations would be a real advantage and likely to be selected for.
I am challenging you to show us grounds for responsible rational freedom on blind chance and mechanical necessity. Absent that freedom there is no “i” there to address a reasoned argument to and all argument is futile.)
Well, since I don't see any decent evidence for consciousness existing outside of our skulls or continuing on after our deaths I'd say I've done that. KF #22
I imagine we all sleep on a regular basis. Being unconscious is not a serious problem for anyone who has dropped a cell phone and seen it then unable to process signals that are indubitably present
When I sleep I dream. When I'm under anaesthetic I have no experiences, no memories. The cell phone analogy is a good one; like it, when I'm 'turned off' I don't process information.ellazimm_{March 23, 2016
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EZ, BTW, I imagine we all sleep on a regular basis. Being unconscious is not a serious problem for anyone who has dropped a cell phone and seen it then unable to process signals that are indubitably present. KFkairosfocus_{March 23, 2016
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PS: Here is Nancy Pearcey:
A major way to test a philosophy or worldview is to ask: Is it logically consistent? Internal contradictions are fatal to any worldview because contradictory statements are necessarily false. “This circle is square” is contradictory, so it has to be false. An especially damaging form of contradiction is self-referential absurdity — which means a theory sets up a definition of truth that it itself fails to meet. Therefore it refutes itself . . . . An example of self-referential absurdity is a theory called evolutionary epistemology, a naturalistic approach that applies evolution to the process of knowing. The theory proposes that the human mind is a product of natural selection. The implication is that the ideas in our minds were selected for their survival value, not for their truth-value. But what if we apply that theory to itself? Then it, too, was selected for survival, not truth — which discredits its own claim to truth. Evolutionary epistemology commits suicide. Astonishingly, many prominent thinkers have embraced the theory without detecting the logical contradiction. Philosopher John Gray writes, “If Darwin’s theory of natural selection is true,… the human mind serves evolutionary success, not truth.” What is the contradiction in that statement? Gray has essentially said, if Darwin’s theory is true, then it “serves evolutionary success, not truth.” In other words, if Darwin’s theory is true, then it is not true. Self-referential absurdity is akin to the well-known liar’s paradox: “This statement is a lie.” If the statement is true, then (as it says) it is not true, but a lie. Another example comes from Francis Crick. In The Astonishing Hypothesis, he writes, “Our highly developed brains, after all, were not evolved under the pressure of discovering scientific truths but only to enable us to be clever enough to survive.” But that means Crick’s own theory is not a “scientific truth.” Applied to itself, the theory commits suicide. Of course, the sheer pressure to survive is likely to produce some correct ideas. A zebra that thinks lions are friendly will not live long. But false ideas may be useful for survival. Evolutionists admit as much: Eric Baum says, “Sometimes you are more likely to survive and propagate if you believe a falsehood than if you believe the truth.” Steven Pinker writes, “Our brains were shaped for fitness, not for truth. Sometimes the truth is adaptive, but sometimes it is not.” The upshot is that survival is no guarantee of truth. If survival is the only standard, we can never know which ideas are true and which are adaptive but false. To make the dilemma even more puzzling, evolutionists tell us that natural selection has produced all sorts of false concepts in the human mind. Many evolutionary materialists maintain that free will is an illusion, consciousness is an illusion, even our sense of self is an illusion — and that all these false ideas were selected for their survival value.
[--> that is, responsible, rational freedom is undermined. Cf here William Provine in his 1998 U Tenn Darwin Day keynote:
Naturalistic evolution has clear consequences that Charles Darwin understood perfectly. 1) No gods worth having exist; 2) no life after death exists; 3) no ultimate foundation for ethics exists; 4) no ultimate meaning in life exists; and 5) human free will is nonexistent . . . . The first 4 implications are so obvious to modern naturalistic evolutionists that I will spend little time defending them. Human free will, however, is another matter. Even evolutionists have trouble swallowing that implication. I will argue that humans are locally determined systems that make choices. They have, however, no free will [--> without responsible freedom, mind, reason and morality alike disintegrate into grand delusion, hence self-referential incoherence and self-refutation. But that does not make such fallacies any less effective in the hands of clever manipulators] . . . [1998 Darwin Day Keynote Address, U of Tenn -- and yes, that is significant i/l/o the Scopes Trial, 1925]

So how can we know whether the theory of evolution itself is one of those false ideas? The theory undercuts itself. A few thinkers, to their credit, recognize the problem. Literary critic Leon Wieseltier writes, “If reason is a product of natural selection, then how much confidence can we have in a rational argument for natural selection? … Evolutionary biology cannot invoke the power of reason even as it destroys it.” On a similar note, philosopher Thomas Nagel asks, “Is the [evolutionary] hypothesis really compatible with the continued confidence in reason as a source of knowledge?” His answer is no: “I have to be able to believe … that I follow the rules of logic because they are correct — not merely because I am biologically programmed to do so.” Hence, “insofar as the evolutionary hypothesis itself depends on reason, it would be self-undermining.” [ENV excerpt, Finding Truth (David C. Cook, 2015) by Nancy Pearcey.]
kairosfocus_{March 23, 2016
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Ez, and, pray tell, am I speaking to an OS cobbled together by blind chance and mechanical necessity? If so, on what grounds do I have any right to expect response to reason or decision and ability to freely follow a reasoned argument and conclude per ground consequent rather than mechanically necessary cause effect with maybe some chance randomness tossed in? (And I do not need to address out of body or out of skull experiences to ask this. Nor am I speaking of what you -- reading between patent lines -- label and dismiss as "religious" texts. I am challenging you to show us grounds for responsible rational freedom on blind chance and mechanical necessity. Absent that freedom there is no "i" there to address a reasoned argument to and all argument is futile.) KFkairosfocus_{March 23, 2016
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KF #16
If we, by contrast, are merely programmed machines we would inevitably be unable to rise above programming, mechanical necessity and/or blind chance.
I respectfully disagree. I find there to be no credible evidence of consciousness existing outside of our skulls. When I see persuasive evidence I'll change my mind. (When I was younger I DID believe in life after death, ESP, etc but when I examined the data I found it lacking.) We don't need to argue about it as I doubt either of us will change our minds. ba77 #17
And just whom might this fictitious “I” be to whom you refer to in your sentence? “that “You”, your joys and your sorrows, your memories and your ambitions, your sense of personal identity and free will, are in fact no more than the behaviour of a vast assembly of nerve cells and their associated molecules. As Lewis Carroll’s Alice might have phrased: “You’re nothing but a pack of neurons.” This hypothesis is so alien to the ideas of most people today that it can truly be called astonishing.”
You find that opinion astonishing, I don't. Have you ever been under a general anaesthetic? I have, twice. While I was 'out' I experienced nothing, felt nothing, no memories whatsoever. I was there then a very brief bit of black and then I was back again but it was several hours later. It was like the intervening time was just snipped out of my experience. I can't imagine how that could happen if there was a part of me that existed outside of my head. Why would I have no record of anything during the time I was 'out' if part of my being was independent of my body? I have heard no credible examples of past life regressions. I have heard no credible examples of out of body experiences. I don't find religious texts to be dependable sources of historical facts. In short I don't believe there is a good reason to believe any part of 'us' lives outside of our brains. As I said to KF, there's no good reason we should discuss it further as we are highly unlikely to change our minds.ellazimm_{March 23, 2016
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“The nitrogen in our DNA, the calcium in our teeth, the iron in our blood, the carbon in our apple pies were made in the interiors of collapsing stars. We are made of starstuff.” — Carl Sagan “Every one of us is, in the cosmic perspective, precious. If a human disagrees with you, let him live. In a hundred billion galaxies, you will not find another.” — Carl Sagan
Zachriel_{March 23, 2016
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as to: "If that’s what it meant then "I" disagree." And just whom might this fictitious "I" be to whom you refer to in your sentence? "that “You”, your joys and your sorrows, your memories and your ambitions, your sense of personal identity and free will, are in fact no more than the behaviour of a vast assembly of nerve cells and their associated molecules. As Lewis Carroll’s Alice might have phrased: “You’re nothing but a pack of neurons.” This hypothesis is so alien to the ideas of most people today that it can truly be called astonishing.” Francis Crick - "The Astonishing Hypothesis" 1994 Atheistic Materialism – Does Richard Dawkins Exist? – video Quote: "It turns out that if every part of you, down to sub-atomic parts, are still what they were when they weren't in you, in other words every ion,,, every single atom that was in the universe,, that has now become part of your living body, is still what is was originally. It hasn't undergone what metaphysicians call a 'substantial change'. So you aren't Richard Dawkins. You are just carbon and neon and sulfur and oxygen and all these individual atoms still. You can spout a philosophy that says scientific materialism, but there aren't any scientific materialists to pronounce it.,,, That's why I think they find it kind of embarrassing to talk that way. Nobody wants to stand up there and say, "You know, I'm not really here". https://www.youtube.com/watch?v=rVCnzq2yTCg&t=37m51s “We have so much confidence in our materialist assumptions (which are assumptions, not facts) that something like free will is denied in principle. Maybe it doesn’t exist, but I don’t really know that. Either way, it doesn’t matter because if free will and consciousness are just an illusion, they are the most seamless illusions ever created. Film maker James Cameron wishes he had special effects that good.” Matthew D. Lieberman – neuroscientist – materialist – UCLA professor And in the following article Dawkins admits that it is impossible to live as if his atheistic worldview were true Who wrote Richard Dawkins's new book? - October 28, 2006 Excerpt: Dawkins: What I do know is that what it feels like to me, and I think to all of us, we don't feel determined. We feel like blaming people for what they do or giving people the credit for what they do. We feel like admiring people for what they do.,,, Manzari: But do you personally see that as an inconsistency in your views? Dawkins: I sort of do. Yes. But it is an inconsistency that we sort of have to live with otherwise life would be intolerable.,,, http://www.evolutionnews.org/2006/10/who_wrote_richard_dawkinss_new002783.html Faith and Science - Dr. Raymond Bohlin - video - (2015) (48:46 minute mark) https://youtu.be/vTIp1kgSqzU?t=2552 At the 23:33 minute mark of the following video, Richard Dawkins agrees with materialistic philosophers who say that: "consciousness is an illusion" A few minutes later Rowan Williams asks Dawkins ”If consciousness is an illusion…what isn’t?”. https://www.youtube.com/watch?v=HWN4cfh1Fac&t=22m57s The Dawkins Delusion - video https://www.youtube.com/watch?v=QERyh9YYEis Sam Harris's Free Will: The Medial Pre-Frontal Cortex Did It - Martin Cothran - November 9, 2012 Excerpt: There is something ironic about the position of thinkers like Harris on issues like this: they claim that their position is the result of the irresistible necessity of logic (in fact, they pride themselves on their logic). Their belief is the consequent, in a ground/consequent relation between their evidence and their conclusion. But their very stated position is that any mental state -- including their position on this issue -- is the effect of a physical, not logical cause. By their own logic, it isn't logic that demands their assent to the claim that free will is an illusion, but the prior chemical state of their brains. The only condition under which we could possibly find their argument convincing is if they are not true. The claim that free will is an illusion requires the possibility that minds have the freedom to assent to a logical argument, a freedom denied by the claim itself. It is an assent that must, in order to remain logical and not physiological, presume a perspective outside the physical order. http://www.evolutionnews.org/2012/11/sam_harriss_fre066221.html (1) rationality implies a thinker in control of thoughts. (2) under materialism a thinker is an effect caused by processes in the brain. (3) in order for materialism to ground rationality a thinker (an effect) must control processes in the brain (a cause). (1)&(2) (4) no effect can control its cause. Therefore materialism cannot ground rationality. per Box UDbornagain77_{March 23, 2016
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EZ, I pause while monitoring a minister's presentation on edu and health etc. Wow, she is pleading with passion just now, DFID is getting an implied lash or two! I think she lost it under provocation and spoke very frankly for a few minutes of truth. Because there is a fundamental issue lurking, I wish to raise a thought or two. It is generally accepted that matter [atoms and the like also energy, distributed in space-time] is governed by laws of chance and/or mechanical necessity. Such are inevitably non-rational matters of cause and effect. Step by step digital algorithm effecting machines and analogue computers alike are not exceptions. Cause-effect bonds of force and mechanical necessity or chance or mixes are inherently non-rational, blind matters of configuration and energy flows. For instance, a ball and disk integrator, an op amp integrator, a NAND logic gate using switches in series, a NOR logic gate using switches in a parallel connexion, the addition of digital feedback through cross-coupled gates to yield an RS latch (the core of storage registers) and so forth. Likewise, feedback loops used for control systems or oscillators. Neural networks, in this context, are coupled amplifier chains with variable links . . . in biosystems, basically pulse repetition rate entities with a fractional change, log response characteristic more or less as the Weber Fechner law speaks of. These machines act by blind forces not by responsible, rational freedom. If we, by contrast, are merely programmed machines we would inevitably be unable to rise above programming, mechanical necessity and/or blind chance. I only point to the issue of imagined materialistic evolutionary origin by cumulative chance and necessity and its inevitable end in self falsification by self referential incoherence: http://iose-gen.blogspot.com/2010/06/origin-of-mind-man-morals-etc.html#slf_ref I emphasise that by contrast, that to reason, understand and know we must be responsibly, rationally free. I point to the Smith Model for an approach to bio cybernetic systems that by way of a two tier controller are open: http://iose-gen.blogspot.com/2010/06/origin-of-mind-man-morals-etc.html#smth_mod Which means that just to be able to discuss like this points beyond evolutionary materialism and its cramped account of mind. KFkairosfocus_{March 23, 2016
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ba77 #7
It is Godel was pointing out that if the human mind were merely material then mathematical intuition would be impossible for humans.
If that's what it meant then I disagree.ellazimm_{March 23, 2016
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I note that in base 101 there infinite primes ending in any of the 101 digits except zero, of which there is exactly one prime that so ends.Roy_{March 23, 2016
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