A341789 -id:A341789 - cp's OEIS frontend

A091727 Norms of prime ideals of Z[sqrt(-5)].

2, 3, 5, 7, 23, 29, 41, 43, 47, 61, 67, 83, 89, 101, 103, 107, 109, 121, 127, 149, 163, 167, 169, 181, 223, 227, 229, 241, 263, 269, 281, 283, 289, 307, 347, 349, 361, 367, 383, 389, 401, 409, 421, 443, 449, 461, 463, 467, 487

Offset: 1

Paul Boddington, Feb 02 2004

Consists of primes congruent to 1, 2, 3, 5, 7, 9 (mod 20) together with the squares of all other primes.
From Jianing Song, Feb 20 2021: (Start)
The norm of a nonzero ideal I in a ring R is defined as the size of the quotient ring R/I.
Note that Z[sqrt(-5)] has class number 2.
For primes p == 1, 9 (mod 20), there are two distinct ideals with norm p in Z[sqrt(-5)], namely (x + y*sqrt(-5)) and (x - y*sqrt(-5)), where (x,y) is a solution to x^2 + 5*y^2 = p.
For p == 3, 7 (mod 20), there are also two distinct ideals with norm p, namely (p, x+y*sqrt(-5)) and (p, x-y*sqrt(-5)), where (x,y) is a solution to x^2 + 5*y^2 = p^2 with y != 0; (2, 1+sqrt(-5)) and (sqrt(-5)) are respectively the unique ideal with norm 2 and 5.
For p == 11, 13, 17, 19 (mod 20), (p) is the only ideal with norm p^2. (End)

			From _Jianing Song_, Feb 20 2021: (Start)
Let |I| be the norm of an ideal I, then:
|(2, 1+sqrt(-5))| = 2;
|(3, 2+sqrt(-5))| = |(3, 2-sqrt(-5))| = 3;
|(sqrt(-5))| = 5;
|(7, 1+3*sqrt(-5))| = |(7, 1-3*sqrt(-5))| = 7;
|(23, 22+3*sqrt(-5))| = |(23, 22-3*sqrt(-5))| = 23;
|(3 + 2*sqrt(-5))| = |(3 - 2*sqrt(-5))| = 29;
|(6 + sqrt(-5))| = |(6 - sqrt(-5))| = 41. (End)

David A. Cox, Primes of the form x^2+ny^2, Wiley, 1989.
A. Frohlich and M. J. Taylor, Algebraic number theory, Cambridge university press, 1991.

Jianing Song, Table of n, a(n) for n = 1..10000

Cf. A091728.
Cf. A289741, A033205, A106865, A139513, A003626.
The number of distinct ideals with norm n is given by A035170.
Norms of prime ideals in O_K, where K is the quadratic field with discriminant D and O_K be the ring of integers of K: A055673 (D=8), A341783 (D=5), A055664 (D=-3), A055025 (D=-4), A090348 (D=-7), A341784 (D=-8), A341785 (D=-11), A341786 (D=-15*), A341787 (D=-19), this sequence (D=-20*), A341788 (D=-43), A341789 (D=-67), A341790 (D=-163). Here a "*" indicates the cases where O_K is not a unique factorization domain.

isA091727(n) = { my(ms = [1, 2, 3, 5, 7, 9], p, e=isprimepower(n,&p)); if(!e || e>2, 0, bitxor(e-1,!!vecsearch(ms,p%20))); }; \\ Antti Karttunen, Feb 24 2020

Offset corrected by Jianing Song, Feb 20 2021

A341784 Norms of prime elements in Z[sqrt(-2)], the ring of integers of Q(sqrt(-2)).

Original entry on oeis.org

2, 3, 11, 17, 19, 25, 41, 43, 49, 59, 67, 73, 83, 89, 97, 107, 113, 131, 137, 139, 163, 169, 179, 193, 211, 227, 233, 241, 251, 257, 281, 283, 307, 313, 331, 337, 347, 353, 379, 401, 409, 419, 433, 443, 449, 457, 467, 491, 499, 521, 523, 529, 547, 563

Offset: 1

Jianing Song, Feb 19 2021

Also norms of prime ideals in Z[sqrt(-2)], which is a unique factorization domain. The norm of a nonzero ideal I in a ring R is defined as the size of the quotient ring R/I.
Consists of the primes congruent to 1, 2, 3 modulo 8 and the squares of primes congruent to 5, 7 modulo 8.
For primes p == 1, 3 (mod 8), there are two distinct ideals with norm p in Z[sqrt(2)], namely (x + y*sqrt(-2)) and (x - y*sqrt(-2)), where (x,y) is a solution to x^2 + 2*y^2 = p; for p = 2, (sqrt(-2)) is the unique ideal with norm p; for p == 5, 7 (mod 8), (p) is the only ideal with norm p^2.

			norm(1 + sqrt(-2)) = norm(1 + sqrt(-2)) = 3;
norm(3 + sqrt(-2)) = norm(3 + sqrt(-2)) = 11;
norm(3 + 2*sqrt(-2)) = norm(3 + 2*sqrt(-2)) = 17;
norm(1 + 3*sqrt(-2)) = norm(1 + 3*sqrt(-2)) = 19.

Jianing Song, Table of n, a(n) for n = 1..10000

Cf. A188510, A033203, A033200, A003628.
The number of nonassociative elements with norm n (also the number of distinct ideals with norm n) is given by A002325.
The total number of elements with norm n is given by A033715.
Norms of prime ideals in O_K, where K is the quadratic field with discriminant D and O_K be the ring of integers of K: A055673 (D=8), A341783 (D=5), A055664 (D=-3), A055025 (D=-4), A090348 (D=-7), this sequence (D=-8), A341785 (D=-11), A341786 (D=-15*), A341787 (D=-19), A091727 (D=-20*), A341788 (D=-43), A341789 (D=-67), A341790 (D=-163). Here a "*" indicates the cases where O_K is not a unique factorization domain.

isA341784(n) = my(disc=-8); (isprime(n) && kronecker(disc,n)>=0) || (issquare(n, &n) && isprime(n) && kronecker(disc,n)==-1)

A341785 Norms of prime elements in Z[(1+sqrt(-11))/2], the ring of integers of Q(sqrt(-11)).

Original entry on oeis.org

3, 4, 5, 11, 23, 31, 37, 47, 49, 53, 59, 67, 71, 89, 97, 103, 113, 137, 157, 163, 169, 179, 181, 191, 199, 223, 229, 251, 257, 269, 289, 311, 313, 317, 331, 353, 361, 367, 379, 383, 389, 397, 401, 419, 421, 433, 443, 449, 463, 467, 487, 499, 509, 521

Offset: 1

Jianing Song, Feb 19 2021

Also norms of prime ideals in Z[(1+sqrt(-11))/2], which is a unique factorization domain. The norm of a nonzero ideal I in a ring R is defined as the size of the quotient ring R/I.
Consists of the primes congruent to 0, 1, 3, 4, 5, 9 modulo 11 and the squares of primes congruent to 2, 6, 7, 8, 10 modulo 5.
For primes p == 1, 3, 4, 5, 9 (mod 11), there are two distinct ideals with norm p in Z[(1+sqrt(-11))/2], namely (x + y*(1+sqrt(-11))/2) and (x + y*(1-sqrt(-11))/2), where (x,y) is a solution to x^2 + x*y + 3*y^2 = p; for p = 11, (sqrt(-11)) is the unique ideal with norm p; for p == 2, 6, 7, 8, 10 (mod 11), (p) is the only ideal with norm p^2.

			norm((1 + sqrt(-11))/2) = norm((1 - sqrt(-11))/2) = 3;
norm((3 + sqrt(-11))/2) = norm((3 - sqrt(-11))/2) = 5;
norm((9 + sqrt(-11))/2) = norm((9 - sqrt(-11))/2) = 23;
norm((5 + 3*sqrt(-11))/2) = norm((5 - 3*sqrt(-11))/2) = 31.

Jianing Song, Table of n, a(n) for n = 1..10000

Cf. A011582, A056874, A296920, A191060.
The number of nonassociative elements with norm n (also the number of distinct ideals with norm n) is given by A035179.
The total number of elements with norm n is given by A028609.
Norms of prime ideals in O_K, where K is the quadratic field with discriminant D and O_K be the ring of integers of K: A055673 (D=8), A341783 (D=5), A055664 (D=-3), A055025 (D=-4), A090348 (D=-7), A341784 (D=-8), this sequence (D=-11), A341786 (D=-15*), A341787 (D=-19), A091727 (D=-20*), A341788 (D=-43), A341789 (D=-67), A341790 (D=-163). Here a "*" indicates the cases where O_K is not a unique factorization domain.

isA341785(n) = my(disc=-11); (isprime(n) && kronecker(disc,n)>=0) || (issquare(n, &n) && isprime(n) && kronecker(disc,n)==-1)

A341786 Norms of prime ideals in Z[(1+sqrt(-15))/2], the ring of integers of Q(sqrt(-15)).

Original entry on oeis.org

2, 3, 5, 17, 19, 23, 31, 47, 49, 53, 61, 79, 83, 107, 109, 113, 121, 137, 139, 151, 167, 169, 173, 181, 197, 199, 211, 227, 229, 233, 241, 257, 263, 271, 293, 317, 331, 347, 349, 353, 379, 383, 409, 421, 439, 443, 467, 499, 503, 541, 557, 563, 571, 587

Offset: 1

Jianing Song, Feb 19 2021

The norm of a nonzero ideal I in a ring R is defined as the size of the quotient ring R/I.
Note that Z[(1+sqrt(-15))/2] has class number 2.
Consists of the primes congruent to 1, 2, 3, 4, 5, 8 modulo 15 and the squares of primes congruent to 7, 11, 13, 14 modulo 15.
For primes p == 1, 4 (mod 15), there are two distinct ideals with norm p in Z[(1+sqrt(-15))/2], namely (x + y*(1+sqrt(-15))/2) and (x + y*(1-sqrt(-15))/2), where (x,y) is a solution to x^2 + x*y + 4*y^2 = p; for p == 2, 8 (mod 15), there are also two distinct ideals with norm p, namely (p, x + y*(1+sqrt(-15))/2) and (p, x + y*(1-sqrt(-15))/2), where (x,y) is a solution to x^2 + x*y + 4*y^2 = p^2 with y != 0; (3, sqrt(-15)) and (5, sqrt(-15)) are respectively the unique ideal with norm 3 and 5; for p == 7, 11, 13, 14 (mod 15), (p) is the only ideal with norm p^2.

			Let |I| be the norm of an ideal I, then:
|(2, (1+sqrt(-15))/2)| = |(2, (1-sqrt(-15))/2)| = 2;
|(3, sqrt(-15))| = 3;
|(5, sqrt(-15))| = 5;
|(17, 7+4*sqrt(-15))| = |(17, 7-4*sqrt(-15))| = 17;
|(2 + sqrt(-15))| = |(2 - sqrt(-15))| = 19;
|(23, 17+4*sqrt(-15))| = |(23, 17-4*sqrt(-15))| = 23;
|(4 + sqrt(-15))| = |(4 - sqrt(-15))| = 31.

Jianing Song, Table of n, a(n) for n = 1..10000

Cf. A316569, A033212, A106859, A191018, A191062.
The number of distinct ideals with norm n is given by A035175.
Norms of prime ideals in O_K, where K is the quadratic field with discriminant D and O_K be the ring of integers of K: A055673 (D=8), A341783 (D=5), A055664 (D=-3), A055025 (D=-4), A090348 (D=-7), A341784 (D=-8), A341785 (D=-11), this sequence (D=-15*), A341787 (D=-19), A091727 (D=-20*), A341788 (D=-43), A341789 (D=-67), A341790 (D=-163). Here a "*" indicates the cases where O_K is not a unique factorization domain.

isA341786(n) = my(disc=-15); (isprime(n) && kronecker(disc,n)>=0) || (issquare(n, &n) && isprime(n) && kronecker(disc,n)==-1)

A341787 Norms of prime elements in Z[(1+sqrt(-19))/2], the ring of integers of Q(sqrt(-19)).

Original entry on oeis.org

4, 5, 7, 9, 11, 17, 19, 23, 43, 47, 61, 73, 83, 101, 131, 137, 139, 149, 157, 163, 169, 191, 197, 199, 229, 233, 239, 251, 263, 271, 277, 283, 311, 313, 347, 349, 353, 359, 367, 389, 397, 419, 443, 457, 461, 463, 467, 479, 491, 499, 503, 541, 557, 571

Offset: 1

Jianing Song, Feb 19 2021

Also norms of prime ideals in Z[(1+sqrt(-19))/2], which is a unique factorization domain. The norm of a nonzero ideal I in a ring R is defined as the size of the quotient ring R/I.
Consists of the primes such that (p,19) >= 0 and the squares of primes such that (p,19) = -1, where (p,19) is the Legendre symbol.
For primes p such that (p,19) = 1, there are two distinct ideals with norm p in Z[(1+sqrt(-19))/2], namely (x + y*(1+sqrt(-19))/2) and (x + y*(1-sqrt(-19))/2), where (x,y) is a solution to x^2 + x*y + 5*y^2 = p; for p = 19, (sqrt(-19)) is the unique ideal with norm p; for primes p with (p,19) = -1, (p) is the only ideal with norm p^2.

			norm((1 + sqrt(-19))/2) = norm((1 - sqrt(-19))/2) = 5;
norm((3 + sqrt(-19))/2) = norm((3 - sqrt(-19))/2) = 7;
norm((5 + sqrt(-19))/2) = norm((5 - sqrt(-19))/2) = 11;
norm((7 + sqrt(-19))/2) = norm((7 - sqrt(-19))/2) = 17.

Jianing Song, Table of n, a(n) for n = 1..10000

Cf. A011585, A106863, A191019, A191063.
The number of nonassociative elements with norm n (also the number of distinct ideals with norm n) is given by A035171.
The total number of elements with norm n is given by A028641.
Norms of prime ideals in O_K, where K is the quadratic field with discriminant D and O_K be the ring of integers of K: A055673 (D=8), A341783 (D=5), A055664 (D=-3), A055025 (D=-4), A090348 (D=-7), A341784 (D=-8), A341785 (D=-11), A341786 (D=-15*), this sequence (D=-19), A091727 (D=-20*), A341788 (D=-43), A341789 (D=-67), A341790 (D=-163). Here a "*" indicates the cases where O_K is not a unique factorization domain.

isA341787(n) = my(disc=-19); (isprime(n) && kronecker(disc,n)>=0) || (issquare(n, &n) && isprime(n) && kronecker(disc,n)==-1)

A341788 Norms of prime elements in Z[(1+sqrt(-43))/2], the ring of integers of Q(sqrt(-43)).

Original entry on oeis.org

4, 9, 11, 13, 17, 23, 25, 31, 41, 43, 47, 49, 53, 59, 67, 79, 83, 97, 101, 103, 107, 109, 127, 139, 167, 173, 181, 193, 197, 229, 239, 251, 269, 271, 281, 283, 293, 307, 311, 317, 337, 353, 359, 361, 367, 379, 397, 401, 431, 439, 443, 461, 479, 487, 509

Offset: 1

Jianing Song, Feb 19 2021

Also norms of prime ideals in Z[(1+sqrt(-43))/2], which is a unique factorization domain. The norm of a nonzero ideal I in a ring R is defined as the size of the quotient ring R/I.
Consists of the primes such that (p,43) >= 0 and the squares of primes such that (p,43) = -1, where (p,43) is the Legendre symbol.
For primes p such that (p,43) = 1, there are two distinct ideals with norm p in Z[(1+sqrt(-43))/2], namely (x + y*(1+sqrt(-43))/2) and (x + y*(1-sqrt(-43))/2), where (x,y) is a solution to x^2 + x*y + 11*y^2 = p; for p = 43, (sqrt(-43)) is the unique ideal with norm p; for primes p with (p,43) = -1, (p) is the only ideal with norm p^2.

			norm((1 + sqrt(-43))/2) = norm((1 - sqrt(-43))/2) = 11;
norm((3 + sqrt(-43))/2) = norm((3 - sqrt(-43))/2) = 13;
norm((5 + sqrt(-43))/2) = norm((5 - sqrt(-43))/2) = 17;
norm((7 + sqrt(-43))/2) = norm((7 - sqrt(-43))/2) = 23;
...
norm((19 + sqrt(-43))/2) = norm((19 - sqrt(-43))/2) = 101.

Jianing Song, Table of n, a(n) for n = 1..10000

Cf. A011591, A106891, A191031, A184902.
The number of nonassociative elements with norm n (also the number of distinct ideals with norm n) is given by A035147.
The total number of elements with norm n is given by A138811.
Norms of prime ideals in O_K, where K is the quadratic field with discriminant D and O_K be the ring of integers of K: A055673 (D=8), A341783 (D=5), A055664 (D=-3), A055025 (D=-4), A090348 (D=-7), A341784 (D=-8), A341785 (D=-11), A341786 (D=-15*), A341787 (D=-19), A091727 (D=-20*), this sequence (D=-43), A341789 (D=-67), A341790 (D=-163). Here a "*" indicates the cases where O_K is not a unique factorization domain.

isA341788(n) = my(disc=-43); (isprime(n) && kronecker(disc,n)>=0) || (issquare(n, &n) && isprime(n) && kronecker(disc,n)==-1)

A341790 Norms of prime elements in Z[(1+sqrt(-163))/2], the ring of integers of Q(sqrt(-163)).

Original entry on oeis.org

4, 9, 25, 41, 43, 47, 49, 53, 61, 71, 83, 97, 113, 121, 131, 151, 163, 167, 169, 173, 179, 197, 199, 223, 227, 251, 263, 281, 289, 307, 313, 347, 359, 361, 367, 373, 379, 383, 397, 409, 419, 421, 439, 457, 461, 487, 499, 503, 523, 529, 547, 563, 577, 593

Offset: 1

Jianing Song, Feb 19 2021

Also norms of prime ideals in Z[(1+sqrt(-163))/2], which is a unique factorization domain. The norm of a nonzero ideal I in a ring R is defined as the size of the quotient ring R/I.
Consists of the primes such that (p,163) >= 0 and the squares of primes such that (p,163) = -1, where (p,163) is the Legendre symbol.
For primes p such that (p,163) = 1, there are two distinct ideals with norm p in Z[(1+sqrt(-163))/2], namely (x + y*(1+sqrt(-163))/2) and (x + y*(1-sqrt(-163))/2), where (x,y) is a solution to x^2 + x*y + 41*y^2 = p; for p = 163, (sqrt(-163)) is the unique ideal with norm p; for primes p with (p,163) = -1, (p) is the only ideal with norm p^2.

			norm((1 + sqrt(-163))/2) = norm((1 - sqrt(-163))/2) = 41;
norm((3 + sqrt(-163))/2) = norm((3 - sqrt(-163))/2) = 43;
norm((5 + sqrt(-163))/2) = norm((5 - sqrt(-163))/2) = 47;
norm((7 + sqrt(-163))/2) = norm((7 - sqrt(-163))/2) = 53;
...
norm((79 + sqrt(-163))/2) = norm((79 - sqrt(-163))/2) = 1601.

Jianing Song, Table of n, a(n) for n = 1..10000

Cf. A011615, A257362, A296921, A296915.
The number of nonassociative elements with norm n (also the number of distinct ideals with norm n) is given by A318983.
The total number of elements with norm n is given by A318985.
Norms of prime ideals in O_K, where K is the quadratic field with discriminant D and O_K be the ring of integers of K: A055673 (D=8), A341783 (D=5), A055664 (D=-3), A055025 (D=-4), A090348 (D=-7), A341784 (D=-8), A341785 (D=-11), A341786 (D=-15*), A341787 (D=-19), A091727 (D=-20*), A341788 (D=-43), A341789 (D=-67), this sequence (D=-163). Here a "*" indicates the cases where O_K is not a unique factorization domain.

isA341783(n) = my(disc=-163); (isprime(n) && kronecker(disc,n)>=0) || (issquare(n, &n) && isprime(n) && kronecker(disc,n)==-1)

A341783 Absolute values of norms of prime elements in Z[(1+sqrt(5))/2], the ring of integers of Q(sqrt(5)).

Original entry on oeis.org

4, 5, 9, 11, 19, 29, 31, 41, 49, 59, 61, 71, 79, 89, 101, 109, 131, 139, 149, 151, 169, 179, 181, 191, 199, 211, 229, 239, 241, 251, 269, 271, 281, 289, 311, 331, 349, 359, 379, 389, 401, 409, 419, 421, 431, 439, 449, 461, 479, 491, 499, 509, 521, 529

Offset: 1

Jianing Song, Feb 19 2021

Also norms of prime ideals in Z[(1+sqrt(5))/2], which is a unique factorization domain. The norm of a nonzero ideal I in a ring R is defined as the size of the quotient ring R/I.
Consists of the primes congruent to 0, 1, 4 modulo 5 and the squares of primes congruent to 2, 3 modulo 5.
For primes p == 1, 4 (mod 5), there are two distinct ideals with norm p in Z[(1+sqrt(5))/2], namely (x + y*(1+sqrt(5))/2) and (x + y*(1-sqrt(5))/2), where (x,y) is a solution to x^2 + x*y - y^2 = p; for p = 5, (sqrt(5)) is the unique ideal with norm p; for p == 2, 3 (mod 5), (p) is the only ideal with norm p^2.

			norm((7 + sqrt(5))/2) = norm((7 - sqrt(5))/2) = 11;
norm((9 + sqrt(5))/2) = norm((9 - sqrt(5))/2) = 19;
norm((11 + sqrt(5))/2) = norm((11 - sqrt(5))/2) = 29;
norm(6 + sqrt(5)) = norm(6 - sqrt(5)) = 31.

Jianing Song, Table of n, a(n) for n = 1..10000

Cf. A080891, A038872, A045468, A003631.
The number of nonassociative elements with absolute value of norm n (also the number of distinct ideals with norm n) is given by A035187.
Norms of prime ideals in O_K, where K is the quadratic field with discriminant D and O_K be the ring of integers of K: A055673 (D=8), this sequence (D=5), A055664 (D=-3), A055025 (D=-4), A090348 (D=-7), A341784 (D=-8), A341785 (D=-11), A341786 (D=-15*), A341787 (D=-19), A091727 (D=-20*), A341788 (D=-43), A341789 (D=-67), A341790 (D=-163). Here a "*" indicates the cases where O_K is not a unique factorization domain.

isA341783(n) = my(disc=5); (isprime(n) && kronecker(disc,n)>=0) || (issquare(n, &n) && isprime(n) && kronecker(disc,n)==-1)

A346725 Use the cells of a hexagonal grid to represent the algebraic integers in the integer ring of Q(sqrt(-67)) as explained in the comments. Number the cells along the counterclockwise hexagonal spiral that starts with cells 0 and 1 representing integers 0 and 1. List the cells that represent 0 or a prime in the ring.

Original entry on oeis.org

0, 2, 3, 5, 6, 7, 8, 9, 11, 13, 14, 15, 17, 19, 20, 21, 22, 23, 25, 26, 28, 29, 30, 31, 32, 34, 35, 37, 39, 41, 43, 45, 47, 49, 51, 53, 55, 57, 59, 61, 62, 63, 64, 65, 66, 67, 68, 69, 71, 72, 73, 74, 76, 77, 78, 79, 80, 81, 82, 83, 84, 86, 87, 88, 89, 91, 95, 97, 101, 103, 107, 109

Offset: 1

Peter Munn, Aug 23 2021

nonn

In this entry we use "rational integers" to refer to integers in their usual sense as whole numbers - they form a subset of the algebraic integers that form the ring, which we denote "R".
The algebraic integers in R (the elements of R) are specifically quadratic integers of the form z = x + y*sqrt(-67) or z = (x+0.5) + (y+0.5)*sqrt(-67) where x and y are rational integers. Plotted as points on a plane, they can be joined in a grid of isosceles triangles or be seen as the center points of hexagonal regions. Adjusting the regions to be regular hexagons makes for appealing diagrams, which we will come to shortly.
(To be precise, we map each element, z, to the region of the complex plane containing the points that have z as their nearest ring element, then map these (hexagonal) regions continuously to the cells of a (regular) hexagonal grid.)
R is one of 9 related rings that are unique factorization domains, meaning their elements factorize into prime elements in a unique way, just as with rational integers and prime numbers. See the Wikipedia link or the Stark reference, for example.
This set of sequences is inspired by tilings: see the Wichmann link. Each tiling represents one of the 9 rings and shows the primes as distinctively colored squares or hexagons as appropriate.
6 other rings (of the 9) can be mapped to the hexagonal grid in the same way. See the comments entitled "General properties of the related hexagonal spiral sequences" in A346721.
This sequence first differs from A346724 at a(70) = 103 <> A346724(70) = 109 and from A346726 at a(116) = 171 <> A346726(116) = 169.
The first rational prime not to appear as a cell number in this sequence is 191.

			The sequence is constructed in the same way as A346721, but the relevant prime is 67 instead of 7. See the example section of A346721.

L. W. Reid, The Elements of the Theory of Algebraic Numbers, MacMillan, NY, 1910.
H. M. Stark, An Introduction to Number Theory. Markham, Chicago, 1970; Theorem 8.22 on page 295 lists the nine UFDs of the form Q(sqrt(-d)), cf. A003173.

OEIS Wiki, Algebraic integers.
Eric Weisstein's World of Mathematics, Complex Plane, Hexagonal Grid, Ring of Integers.
Brian Wichmann, Tiling for Unique Factorization Domains, Jul 22 2019. See Figure 9.
Wikipedia, Quadratic integer.
Wikipedia, Unique factorization domain.

Cf. A003173, A345764.
Norms of primes in R: A341789.
Equivalent sequences for other Q(sqrt(D)): A345436 (D=-1), A345437 (D=-2), A345435 (D=-3), A346721 (D=-7), A346722 (D=-11), A346723 (D=-19), A346724 (D=-43), A346726 (D=-163).

m is a term if and only if A345764(m) is a term.

cp's OEIS Frontend

A091727 Norms of prime ideals of Z[sqrt(-5)].

Views

Author

Keywords

Comments

Examples

References

Links

Crossrefs

Programs

PARI

Extensions

A341784 Norms of prime elements in Z[sqrt(-2)], the ring of integers of Q(sqrt(-2)).

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

PARI

A341785 Norms of prime elements in Z[(1+sqrt(-11))/2], the ring of integers of Q(sqrt(-11)).

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

PARI

A341786 Norms of prime ideals in Z[(1+sqrt(-15))/2], the ring of integers of Q(sqrt(-15)).

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

PARI

A341787 Norms of prime elements in Z[(1+sqrt(-19))/2], the ring of integers of Q(sqrt(-19)).

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

PARI

A341788 Norms of prime elements in Z[(1+sqrt(-43))/2], the ring of integers of Q(sqrt(-43)).

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

PARI

A341790 Norms of prime elements in Z[(1+sqrt(-163))/2], the ring of integers of Q(sqrt(-163)).

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

PARI

A341783 Absolute values of norms of prime elements in Z[(1+sqrt(5))/2], the ring of integers of Q(sqrt(5)).

Views

Author

Keywords

Comments

Examples

Links