Primes without a 7 - Numberphile
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Intuitively, I thought that there must be less and less numbers without 7 by increasing size, but it helps, if I look at the problem from the other way, from a constructional way.
The wrong and misleading idea by seeing such a long and long and long crafted number, like the one in the book, that it looks like a random number, but the smallest change in it might make it a non-prime. And the longer the number gets, the more numbers exist of equal length.
For a 7-digit number like 1. 000. 000, there are 10-7-10-6 numbers, being of the same length.
That is about 900. 000 numbers. Now, how many of these numbers are written without a 7?
Well - it's easy, 9-7-9-6, isn't it?
With a smaller exponent: There are 90 numbers of length 2 (10, 11, , 99) and 72 of them contain no 7. There are 9 7s in the pattern 17, 27, 37, . and 10 of the form 70, 71, . 79, but 77 is counted twice, so 18 2-digit numbers which contain at least one 7.
So there are 4. 251. 528 (continental dot) numbers without 7 from 1. 000. 000 to 9. 999. 999 - it's not too astonishing, that some of them are primes.
There are 1M primes below 15. 485. 864 and 416. 913 of them contain no 7.
Hey, that reminds me weakly of a project Euler challenge - I have to find that question, to try if I can answer it with my little insight.
Date: 2022-04-09
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Comments and reviews: 9
TheAbc45678
Here's the frequency distribution of integers in his paper: -0: 1269, 1: 3714, 2: 2177, 3: 733, 4: 312, 5: 321, 6: 274, 7: 395, 8: 143, 9: 242, 10: 427, 11: 54, 12: 58, 13: 65-. I was expecting 7 to be higher, but it wasn't. Then I realized the paper wasn't about the number 7 even though the video (and its title) gave me that impression. Bear in mind that these frequencies include dates, page numbers, section numbers, etc. If I had a copy of the paper in an editable format I would have edited those things out before doing the counts. It's sad that an editable copy of this paper isn't available online. Although the paper isn't about -7- it should, and does, contain a spike for the integer -10- because this research is done in base 10.
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Here's the frequency distribution of integers in his paper: -0: 1269, 1: 3714, 2: 2177, 3: 733, 4: 312, 5: 321, 6: 274, 7: 395, 8: 143, 9: 242, 10: 427, 11: 54, 12: 58, 13: 65-. I was expecting 7 to be higher, but it wasn't. Then I realized the paper wasn't about the number 7 even though the video (and its title) gave me that impression. Bear in mind that these frequencies include dates, page numbers, section numbers, etc. If I had a copy of the paper in an editable format I would have edited those things out before doing the counts. It's sad that an editable copy of this paper isn't available online. Although the paper isn't about -7- it should, and does, contain a spike for the integer -10- because this research is done in base 10.
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spectronix1
Perhaps another interesting number system would be a mixed-radix number system, one that is associated to a residue number system containing a modulus for each of the first n primes, where the product of the first n primes sets the range of the number system. In other words, to use an RNS or mixed-radix number representation for this question, one needs to define the rule for the automatic extension of the number system to infinity because the -radix- is not fixed. This is a very intuitive rule for a more advanced look into the properties of digits versus primes as it is fundamental, not arbitrary as is fixed-radix. There's a lot more questions to ask. Maitrix.
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Perhaps another interesting number system would be a mixed-radix number system, one that is associated to a residue number system containing a modulus for each of the first n primes, where the product of the first n primes sets the range of the number system. In other words, to use an RNS or mixed-radix number representation for this question, one needs to define the rule for the automatic extension of the number system to infinity because the -radix- is not fixed. This is a very intuitive rule for a more advanced look into the properties of digits versus primes as it is fundamental, not arbitrary as is fixed-radix. There's a lot more questions to ask. Maitrix.
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Glenn
So I started playing a bit with primes today and saw that all prime numbers I could find ended on a 1 when written in binary. A quick google search showed that I was not the first to discover this. But I was wondering if there is a reason for this, if it really is true for ALL primes, and whether it has any complications for computer security since (from what I understand) most of encryption -stuff- uses primes to do its thing. (Novice in both math and computing, but I have a general interest.
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So I started playing a bit with primes today and saw that all prime numbers I could find ended on a 1 when written in binary. A quick google search showed that I was not the first to discover this. But I was wondering if there is a reason for this, if it really is true for ALL primes, and whether it has any complications for computer security since (from what I understand) most of encryption -stuff- uses primes to do its thing. (Novice in both math and computing, but I have a general interest.
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Kevin
Doesn't the existence of primes made of repeating 1s or repeating 7s prove that there exists primes with none of every digit? A prime of repeating 1s obviously contains no 0s, 2s, 3s, 4s, 5s, 6s, 7s, 8s, or 9s while a prime of repeating 7s obviously contains no 1s. Wouldn't you simply need to prove there exists an infinite number of primes of repeating 1s and 7s to have the answer you're looking for?
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Doesn't the existence of primes made of repeating 1s or repeating 7s prove that there exists primes with none of every digit? A prime of repeating 1s obviously contains no 0s, 2s, 3s, 4s, 5s, 6s, 7s, 8s, or 9s while a prime of repeating 7s obviously contains no 1s. Wouldn't you simply need to prove there exists an infinite number of primes of repeating 1s and 7s to have the answer you're looking for?
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kurzackd
Pardon me, but wouldn't the most logical way to approach this problem be to analyze the primes that have -already- been found up to the current largest prime found?
i. e. go through each prime found and start analyzing patterns in present and missing digits, frequency distributions, et cetera.
Wouldn't that be the most sensible way to go?
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Pardon me, but wouldn't the most logical way to approach this problem be to analyze the primes that have -already- been found up to the current largest prime found?
i. e. go through each prime found and start analyzing patterns in present and missing digits, frequency distributions, et cetera.
Wouldn't that be the most sensible way to go?
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SomeoneCommenting
The first thing that you need to know before tackling these problems or reading papers about them, is to understand all those hieroglyphs used by mathematicians to establish their equations. For us simple mortals, I bet that even if they explained it all only with words, I still wouldn't understand a thing.
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The first thing that you need to know before tackling these problems or reading papers about them, is to understand all those hieroglyphs used by mathematicians to establish their equations. For us simple mortals, I bet that even if they explained it all only with words, I still wouldn't understand a thing.
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Jeff
This is a subset of much stronger proposition: are the primes as a digit sequence normal? Is the set of all digit sequences found in a prime number the same as the set of all possible digit sequences? Looking at what can and can't be proved here shows how far we are from a proof of what seems intuitively obvious.
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This is a subset of much stronger proposition: are the primes as a digit sequence normal? Is the set of all digit sequences found in a prime number the same as the set of all possible digit sequences? Looking at what can and can't be proved here shows how far we are from a proof of what seems intuitively obvious.
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r0kus
I don't understand the point of this limited exploration. It is simply one aspect of base-10 numeric representation. If the base is 7 or below, there will never be any 7 numerals. Or if the base is shifted to 11, 16, 36, etc, the incidence of any given numeral is different.
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I don't understand the point of this limited exploration. It is simply one aspect of base-10 numeric representation. If the base is 7 or below, there will never be any 7 numerals. Or if the base is shifted to 11, 16, 36, etc, the incidence of any given numeral is different.
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Brandon
Wouldnt a number made of all 7's be divisible by 7 making it not prime? I would imagine only some numbers made entirely of 1's would be prime. Something like 2/3 of them, maybe? Assuming that they aren't divisible by a number ending in 3 & another number ending in 7?
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Wouldnt a number made of all 7's be divisible by 7 making it not prime? I would imagine only some numbers made entirely of 1's would be prime. Something like 2/3 of them, maybe? Assuming that they aren't divisible by a number ending in 3 & another number ending in 7?
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