Distributions of prime numbers
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When folding the number line not into a spiral (like Ulam did) but into a zig-zag pattern (like Cantor did) there are other patterns visible in the distribution of prime numbers:
[To see these pictures in higher resolution, you can try it out here and here. Note that these are two different zig-zag patterns (with different "lattice parameters").]
One sees diagonal bands with an above-average density of prime numbers. One of them contains the prime numbers $7, 11, 29, 43, 73, 89, 131, 157, 181, 211, 239, 379, 421, 461, 601,dots $.
How can these bands be explained?
One also sees a rather distinct band from the lower left going upwards at an angle of roughly $75°$. It contains the prime numbers $229, 271, 317, 367, 421, 479, 541, 607, 677, 751, dots $.
Note, that the vertical and horizontal borders of the pattern contain no primes at all, which can be easily understood.
[See my related question concerning the distributions of other numbers.]
By the way (and off the record): The detecting of lines in patterns (most easily of straight lines) is fun also in non-mathematical contexts (be them obvious or not):
Giotto (ca. 1267 – 1337) Entombment of Mary
prime-numbers arithmetic visualization experimental-mathematics
asked Aug 17 at 9:10
Hans Stricker
4,33813574
add a comment |Â
up vote
5
down vote
favorite
When folding the number line not into a spiral (like Ulam did) but into a zig-zag pattern (like Cantor did) there are other patterns visible in the distribution of prime numbers:
[To see these pictures in higher resolution, you can try it out here and here. Note that these are two different zig-zag patterns (with different "lattice parameters").]
One sees diagonal bands with an above-average density of prime numbers. One of them contains the prime numbers $7, 11, 29, 43, 73, 89, 131, 157, 181, 211, 239, 379, 421, 461, 601,dots $.
How can these bands be explained?
One also sees a rather distinct band from the lower left going upwards at an angle of roughly $75°$. It contains the prime numbers $229, 271, 317, 367, 421, 479, 541, 607, 677, 751, dots $.
Note, that the vertical and horizontal borders of the pattern contain no primes at all, which can be easily understood.
[See my related question concerning the distributions of other numbers.]
By the way (and off the record): The detecting of lines in patterns (most easily of straight lines) is fun also in non-mathematical contexts (be them obvious or not):
Giotto (ca. 1267 – 1337) Entombment of Mary
prime-numbers arithmetic visualization experimental-mathematics
asked Aug 17 at 9:10
Hans Stricker
4,33813574
3
I think it's still more or less the same effect that the Ulam spiral illustrates: that some quadratic expressions generate more primes than one might expect.
– Arthur
Aug 17 at 9:26
Yes, in principal, more or less. But Ulam-like coverings of $mathbbZtimes mathbbZ$ (spirals) may exhibit higher level spirals (see here), while Cantor-like coverings of $mathbbNtimes mathbbN$ may exhibit straight lines along arbitrary angles (as can be seen above). Both long for different explanations I guess (see Aarons explanations here).
– Hans Stricker
Aug 17 at 17:11
1
The first sequence you described corresponds to A081352 (polynomials $n^2+3npm 1$), while second seems to be contained in A007641 (polynomial $2n^2+29)$. Maybe it won't be too hard to show that each line will corresponds to one or more such polynomials.
– Sil
Aug 19 at 17:14
1
The Bunyakovsky conjecture is related (generalization of Dirichlet's theorem to generic degree polynomials)
– Sil
Aug 19 at 17:45
1
Ah, A081352 is even called "Main diagonal of square maze arrangement of natural numbers". The question is: Why does already the main diagonal produce primes? The other way around: How probable is it that a polynomial of degree 2 produces more primes than expected?
– Hans Stricker
Aug 19 at 17:55
add a comment |Â
up vote
5
down vote
favorite
up vote
5
down vote
favorite
When folding the number line not into a spiral (like Ulam did) but into a zig-zag pattern (like Cantor did) there are other patterns visible in the distribution of prime numbers:
[To see these pictures in higher resolution, you can try it out here and here. Note that these are two different zig-zag patterns (with different "lattice parameters").]
One sees diagonal bands with an above-average density of prime numbers. One of them contains the prime numbers $7, 11, 29, 43, 73, 89, 131, 157, 181, 211, 239, 379, 421, 461, 601,dots $.
How can these bands be explained?
One also sees a rather distinct band from the lower left going upwards at an angle of roughly $75°$. It contains the prime numbers $229, 271, 317, 367, 421, 479, 541, 607, 677, 751, dots $.
Note, that the vertical and horizontal borders of the pattern contain no primes at all, which can be easily understood.
[See my related question concerning the distributions of other numbers.]
By the way (and off the record): The detecting of lines in patterns (most easily of straight lines) is fun also in non-mathematical contexts (be them obvious or not):
Giotto (ca. 1267 – 1337) Entombment of Mary
prime-numbers arithmetic visualization experimental-mathematics
asked Aug 17 at 9:10
Hans Stricker
4,33813574
When folding the number line not into a spiral (like Ulam did) but into a zig-zag pattern (like Cantor did) there are other patterns visible in the distribution of prime numbers:
[To see these pictures in higher resolution, you can try it out here and here. Note that these are two different zig-zag patterns (with different "lattice parameters").]
One sees diagonal bands with an above-average density of prime numbers. One of them contains the prime numbers $7, 11, 29, 43, 73, 89, 131, 157, 181, 211, 239, 379, 421, 461, 601,dots $.
How can these bands be explained?
One also sees a rather distinct band from the lower left going upwards at an angle of roughly $75°$. It contains the prime numbers $229, 271, 317, 367, 421, 479, 541, 607, 677, 751, dots $.
Note, that the vertical and horizontal borders of the pattern contain no primes at all, which can be easily understood.
[See my related question concerning the distributions of other numbers.]
By the way (and off the record): The detecting of lines in patterns (most easily of straight lines) is fun also in non-mathematical contexts (be them obvious or not):
Giotto (ca. 1267 – 1337) Entombment of Mary
prime-numbers arithmetic visualization experimental-mathematics
asked Aug 17 at 9:10
Hans Stricker
4,33813574
edited Aug 19 at 16:58
asked Aug 17 at 9:10
Hans Stricker
4,33813574
asked Aug 17 at 9:10
Hans Stricker
4,33813574
asked Aug 17 at 9:10
Hans Stricker
4,33813574
4,33813574
3
I think it's still more or less the same effect that the Ulam spiral illustrates: that some quadratic expressions generate more primes than one might expect.
– Arthur
Aug 17 at 9:26
Yes, in principal, more or less. But Ulam-like coverings of $mathbbZtimes mathbbZ$ (spirals) may exhibit higher level spirals (see here), while Cantor-like coverings of $mathbbNtimes mathbbN$ may exhibit straight lines along arbitrary angles (as can be seen above). Both long for different explanations I guess (see Aarons explanations here).
– Hans Stricker
Aug 17 at 17:11
1
The first sequence you described corresponds to A081352 (polynomials $n^2+3npm 1$), while second seems to be contained in A007641 (polynomial $2n^2+29)$. Maybe it won't be too hard to show that each line will corresponds to one or more such polynomials.
– Sil
Aug 19 at 17:14
1
The Bunyakovsky conjecture is related (generalization of Dirichlet's theorem to generic degree polynomials)
– Sil
Aug 19 at 17:45
1
Ah, A081352 is even called "Main diagonal of square maze arrangement of natural numbers". The question is: Why does already the main diagonal produce primes? The other way around: How probable is it that a polynomial of degree 2 produces more primes than expected?
– Hans Stricker
Aug 19 at 17:55
add a comment |Â
3
I think it's still more or less the same effect that the Ulam spiral illustrates: that some quadratic expressions generate more primes than one might expect.
– Arthur
Aug 17 at 9:26
Yes, in principal, more or less. But Ulam-like coverings of $mathbbZtimes mathbbZ$ (spirals) may exhibit higher level spirals (see here), while Cantor-like coverings of $mathbbNtimes mathbbN$ may exhibit straight lines along arbitrary angles (as can be seen above). Both long for different explanations I guess (see Aarons explanations here).
– Hans Stricker
Aug 17 at 17:11
1
The first sequence you described corresponds to A081352 (polynomials $n^2+3npm 1$), while second seems to be contained in A007641 (polynomial $2n^2+29)$. Maybe it won't be too hard to show that each line will corresponds to one or more such polynomials.
– Sil
Aug 19 at 17:14
1
The Bunyakovsky conjecture is related (generalization of Dirichlet's theorem to generic degree polynomials)
– Sil
Aug 19 at 17:45
1
Ah, A081352 is even called "Main diagonal of square maze arrangement of natural numbers". The question is: Why does already the main diagonal produce primes? The other way around: How probable is it that a polynomial of degree 2 produces more primes than expected?
– Hans Stricker
Aug 19 at 17:55
3
3
I think it's still more or less the same effect that the Ulam spiral illustrates: that some quadratic expressions generate more primes than one might expect.
– Arthur
Aug 17 at 9:26
I think it's still more or less the same effect that the Ulam spiral illustrates: that some quadratic expressions generate more primes than one might expect.
– Arthur
Aug 17 at 9:26
Yes, in principal, more or less. But Ulam-like coverings of $mathbbZtimes mathbbZ$ (spirals) may exhibit higher level spirals (see here), while Cantor-like coverings of $mathbbNtimes mathbbN$ may exhibit straight lines along arbitrary angles (as can be seen above). Both long for different explanations I guess (see Aarons explanations here).
– Hans Stricker
Aug 17 at 17:11
Yes, in principal, more or less. But Ulam-like coverings of $mathbbZtimes mathbbZ$ (spirals) may exhibit higher level spirals (see here), while Cantor-like coverings of $mathbbNtimes mathbbN$ may exhibit straight lines along arbitrary angles (as can be seen above). Both long for different explanations I guess (see Aarons explanations here).
– Hans Stricker
Aug 17 at 17:11
1
1
The first sequence you described corresponds to A081352 (polynomials $n^2+3npm 1$), while second seems to be contained in A007641 (polynomial $2n^2+29)$. Maybe it won't be too hard to show that each line will corresponds to one or more such polynomials.
– Sil
Aug 19 at 17:14
The first sequence you described corresponds to A081352 (polynomials $n^2+3npm 1$), while second seems to be contained in A007641 (polynomial $2n^2+29)$. Maybe it won't be too hard to show that each line will corresponds to one or more such polynomials.
– Sil
Aug 19 at 17:14
1
1
The Bunyakovsky conjecture is related (generalization of Dirichlet's theorem to generic degree polynomials)
– Sil
Aug 19 at 17:45
The Bunyakovsky conjecture is related (generalization of Dirichlet's theorem to generic degree polynomials)
– Sil
Aug 19 at 17:45
1
1
Ah, A081352 is even called "Main diagonal of square maze arrangement of natural numbers". The question is: Why does already the main diagonal produce primes? The other way around: How probable is it that a polynomial of degree 2 produces more primes than expected?
– Hans Stricker
Aug 19 at 17:55
Ah, A081352 is even called "Main diagonal of square maze arrangement of natural numbers". The question is: Why does already the main diagonal produce primes? The other way around: How probable is it that a polynomial of degree 2 produces more primes than expected?
– Hans Stricker
Aug 19 at 17:55
add a comment |Â
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3
I think it's still more or less the same effect that the Ulam spiral illustrates: that some quadratic expressions generate more primes than one might expect.
– Arthur
Aug 17 at 9:26
Yes, in principal, more or less. But Ulam-like coverings of $mathbbZtimes mathbbZ$ (spirals) may exhibit higher level spirals (see here), while Cantor-like coverings of $mathbbNtimes mathbbN$ may exhibit straight lines along arbitrary angles (as can be seen above). Both long for different explanations I guess (see Aarons explanations here).
– Hans Stricker
Aug 17 at 17:11
1
The first sequence you described corresponds to A081352 (polynomials $n^2+3npm 1$), while second seems to be contained in A007641 (polynomial $2n^2+29)$. Maybe it won't be too hard to show that each line will corresponds to one or more such polynomials.
– Sil
Aug 19 at 17:14
1
The Bunyakovsky conjecture is related (generalization of Dirichlet's theorem to generic degree polynomials)
– Sil
Aug 19 at 17:45
1
Ah, A081352 is even called "Main diagonal of square maze arrangement of natural numbers". The question is: Why does already the main diagonal produce primes? The other way around: How probable is it that a polynomial of degree 2 produces more primes than expected?
– Hans Stricker
Aug 19 at 17:55