Almost- and non-primes in the Ulam spiral
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There are lots of explanations of the preferred diagonal distribution of prime numbers in the Ulam spiral:
I wonder if these explanations can also explain the observation that when highlighting also "almost-primes" – i.e. numbers $m$ with large enough totient function $varphi(m)$ (for prime numbers $p$ we have $varphi(p) = p-1$) – a checkerboard pattern emerges:
Note that the values of $(varphi(m)+1)/m$ are indicated by different shades of gray, and only numbers $m$ with $(varphi(m)+1)/m geq 0.52$ are highlighted.
Note also, that the only numbers that disturb the visible checkerboard are
false positive: $2,4,8,16,32$ (but not $64, 128, 256$)
false negative: $105,165,195,255,285$
This is, how the spiral looks like when all numbers are displayed:
Note that – of course – also numbers $m$ with a small value of the totient function $varphi(m)$ ("almost non-primes") tend to be arranged along diagonals:
(Only $m$ with $(varphi(m)+1)/m leq 0.48$ are highlighted.)
elementary-number-theory prime-numbers visualization
asked Jan 8 at 16:21
Hans StrickerHans Stricker
6,31843990
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add a comment |
$begingroup$
There are lots of explanations of the preferred diagonal distribution of prime numbers in the Ulam spiral:
I wonder if these explanations can also explain the observation that when highlighting also "almost-primes" – i.e. numbers $m$ with large enough totient function $varphi(m)$ (for prime numbers $p$ we have $varphi(p) = p-1$) – a checkerboard pattern emerges:
Note that the values of $(varphi(m)+1)/m$ are indicated by different shades of gray, and only numbers $m$ with $(varphi(m)+1)/m geq 0.52$ are highlighted.
Note also, that the only numbers that disturb the visible checkerboard are
false positive: $2,4,8,16,32$ (but not $64, 128, 256$)
false negative: $105,165,195,255,285$
This is, how the spiral looks like when all numbers are displayed:
Note that – of course – also numbers $m$ with a small value of the totient function $varphi(m)$ ("almost non-primes") tend to be arranged along diagonals:
(Only $m$ with $(varphi(m)+1)/m leq 0.48$ are highlighted.)
elementary-number-theory prime-numbers visualization
asked Jan 8 at 16:21
Hans StrickerHans Stricker
6,31843990
$endgroup$
1
$begingroup$
My first observation when looking at your so-called "checkerboard pattern", is that you've mostly colored the odd numbers, which obviously give that pattern.
$endgroup$
– Henrik
Jan 8 at 16:31
1
$begingroup$
Ulam spiral seems to me to be the search for patterns that do not actually exist. The main issue is : How do we even define "pattern" ?
$endgroup$
– Peter
Jan 8 at 16:38
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@Henrik: Thanks a lot for this hint. It gave rise to this follow-up question - let's see if an answer is given there.
$endgroup$
– Hans Stricker
Jan 9 at 10:23
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@Peter: I disagree. Patterns do exist and they are meaningful, as can be possibly seen here
$endgroup$
– Hans Stricker
Jan 9 at 10:24
$begingroup$
What about the definition of a "pattern" ? Can it be defined rigorously ? If not, we cannot claim that the prime numbers exhibit a pattern. We can find a (local!) structure that APPEARS to be a pattern, but when we go to larger primes, this pattern will almost surely vanish anyway.
$endgroup$
– Peter
Jan 9 at 10:37
add a comment |
$begingroup$
There are lots of explanations of the preferred diagonal distribution of prime numbers in the Ulam spiral:
I wonder if these explanations can also explain the observation that when highlighting also "almost-primes" – i.e. numbers $m$ with large enough totient function $varphi(m)$ (for prime numbers $p$ we have $varphi(p) = p-1$) – a checkerboard pattern emerges:
Note that the values of $(varphi(m)+1)/m$ are indicated by different shades of gray, and only numbers $m$ with $(varphi(m)+1)/m geq 0.52$ are highlighted.
Note also, that the only numbers that disturb the visible checkerboard are
false positive: $2,4,8,16,32$ (but not $64, 128, 256$)
false negative: $105,165,195,255,285$
This is, how the spiral looks like when all numbers are displayed:
Note that – of course – also numbers $m$ with a small value of the totient function $varphi(m)$ ("almost non-primes") tend to be arranged along diagonals:
(Only $m$ with $(varphi(m)+1)/m leq 0.48$ are highlighted.)
elementary-number-theory prime-numbers visualization
asked Jan 8 at 16:21
Hans StrickerHans Stricker
6,31843990
$endgroup$
There are lots of explanations of the preferred diagonal distribution of prime numbers in the Ulam spiral:
I wonder if these explanations can also explain the observation that when highlighting also "almost-primes" – i.e. numbers $m$ with large enough totient function $varphi(m)$ (for prime numbers $p$ we have $varphi(p) = p-1$) – a checkerboard pattern emerges:
Note that the values of $(varphi(m)+1)/m$ are indicated by different shades of gray, and only numbers $m$ with $(varphi(m)+1)/m geq 0.52$ are highlighted.
Note also, that the only numbers that disturb the visible checkerboard are
false positive: $2,4,8,16,32$ (but not $64, 128, 256$)
false negative: $105,165,195,255,285$
This is, how the spiral looks like when all numbers are displayed:
Note that – of course – also numbers $m$ with a small value of the totient function $varphi(m)$ ("almost non-primes") tend to be arranged along diagonals:
(Only $m$ with $(varphi(m)+1)/m leq 0.48$ are highlighted.)
elementary-number-theory prime-numbers visualization
elementary-number-theory prime-numbers visualization
asked Jan 8 at 16:21
Hans StrickerHans Stricker
6,31843990
asked Jan 8 at 16:21
Hans StrickerHans Stricker
6,31843990
edited Jan 8 at 16:33
Hans Stricker
asked Jan 8 at 16:21
Hans StrickerHans Stricker
6,31843990
asked Jan 8 at 16:21
Hans StrickerHans Stricker
6,31843990
asked Jan 8 at 16:21
Hans StrickerHans Stricker
6,31843990
6,31843990
1
$begingroup$
My first observation when looking at your so-called "checkerboard pattern", is that you've mostly colored the odd numbers, which obviously give that pattern.
$endgroup$
– Henrik
Jan 8 at 16:31
1
$begingroup$
Ulam spiral seems to me to be the search for patterns that do not actually exist. The main issue is : How do we even define "pattern" ?
$endgroup$
– Peter
Jan 8 at 16:38
$begingroup$
@Henrik: Thanks a lot for this hint. It gave rise to this follow-up question - let's see if an answer is given there.
$endgroup$
– Hans Stricker
Jan 9 at 10:23
$begingroup$
@Peter: I disagree. Patterns do exist and they are meaningful, as can be possibly seen here
$endgroup$
– Hans Stricker
Jan 9 at 10:24
$begingroup$
What about the definition of a "pattern" ? Can it be defined rigorously ? If not, we cannot claim that the prime numbers exhibit a pattern. We can find a (local!) structure that APPEARS to be a pattern, but when we go to larger primes, this pattern will almost surely vanish anyway.
$endgroup$
– Peter
Jan 9 at 10:37
add a comment |
1
$begingroup$
My first observation when looking at your so-called "checkerboard pattern", is that you've mostly colored the odd numbers, which obviously give that pattern.
$endgroup$
– Henrik
Jan 8 at 16:31
1
$begingroup$
Ulam spiral seems to me to be the search for patterns that do not actually exist. The main issue is : How do we even define "pattern" ?
$endgroup$
– Peter
Jan 8 at 16:38
$begingroup$
@Henrik: Thanks a lot for this hint. It gave rise to this follow-up question - let's see if an answer is given there.
$endgroup$
– Hans Stricker
Jan 9 at 10:23
$begingroup$
@Peter: I disagree. Patterns do exist and they are meaningful, as can be possibly seen here
$endgroup$
– Hans Stricker
Jan 9 at 10:24
$begingroup$
What about the definition of a "pattern" ? Can it be defined rigorously ? If not, we cannot claim that the prime numbers exhibit a pattern. We can find a (local!) structure that APPEARS to be a pattern, but when we go to larger primes, this pattern will almost surely vanish anyway.
$endgroup$
– Peter
Jan 9 at 10:37
1
1
$begingroup$
My first observation when looking at your so-called "checkerboard pattern", is that you've mostly colored the odd numbers, which obviously give that pattern.
$endgroup$
– Henrik
Jan 8 at 16:31
$begingroup$
My first observation when looking at your so-called "checkerboard pattern", is that you've mostly colored the odd numbers, which obviously give that pattern.
$endgroup$
– Henrik
Jan 8 at 16:31
1
1
$begingroup$
Ulam spiral seems to me to be the search for patterns that do not actually exist. The main issue is : How do we even define "pattern" ?
$endgroup$
– Peter
Jan 8 at 16:38
$begingroup$
Ulam spiral seems to me to be the search for patterns that do not actually exist. The main issue is : How do we even define "pattern" ?
$endgroup$
– Peter
Jan 8 at 16:38
$begingroup$
@Henrik: Thanks a lot for this hint. It gave rise to this follow-up question - let's see if an answer is given there.
$endgroup$
– Hans Stricker
Jan 9 at 10:23
$begingroup$
@Henrik: Thanks a lot for this hint. It gave rise to this follow-up question - let's see if an answer is given there.
$endgroup$
– Hans Stricker
Jan 9 at 10:23
$begingroup$
@Peter: I disagree. Patterns do exist and they are meaningful, as can be possibly seen here
$endgroup$
– Hans Stricker
Jan 9 at 10:24
$begingroup$
@Peter: I disagree. Patterns do exist and they are meaningful, as can be possibly seen here
$endgroup$
– Hans Stricker
Jan 9 at 10:24
$begingroup$
What about the definition of a "pattern" ? Can it be defined rigorously ? If not, we cannot claim that the prime numbers exhibit a pattern. We can find a (local!) structure that APPEARS to be a pattern, but when we go to larger primes, this pattern will almost surely vanish anyway.
$endgroup$
– Peter
Jan 9 at 10:37
$begingroup$
What about the definition of a "pattern" ? Can it be defined rigorously ? If not, we cannot claim that the prime numbers exhibit a pattern. We can find a (local!) structure that APPEARS to be a pattern, but when we go to larger primes, this pattern will almost surely vanish anyway.
$endgroup$
– Peter
Jan 9 at 10:37
add a comment |
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$begingroup$
My first observation when looking at your so-called "checkerboard pattern", is that you've mostly colored the odd numbers, which obviously give that pattern.
$endgroup$
– Henrik
Jan 8 at 16:31
1
$begingroup$
Ulam spiral seems to me to be the search for patterns that do not actually exist. The main issue is : How do we even define "pattern" ?
$endgroup$
– Peter
Jan 8 at 16:38
$begingroup$
@Henrik: Thanks a lot for this hint. It gave rise to this follow-up question - let's see if an answer is given there.
$endgroup$
– Hans Stricker
Jan 9 at 10:23
$begingroup$
@Peter: I disagree. Patterns do exist and they are meaningful, as can be possibly seen here
$endgroup$
– Hans Stricker
Jan 9 at 10:24
$begingroup$
What about the definition of a "pattern" ? Can it be defined rigorously ? If not, we cannot claim that the prime numbers exhibit a pattern. We can find a (local!) structure that APPEARS to be a pattern, but when we go to larger primes, this pattern will almost surely vanish anyway.
$endgroup$
– Peter
Jan 9 at 10:37