A324024 -id:A324024 - cp's OEIS frontend

A324026 Digits of one of the two 5-adic integers sqrt(6) that is related to A324024.

4, 1, 4, 0, 2, 3, 2, 1, 3, 1, 1, 4, 1, 1, 2, 1, 2, 2, 2, 0, 1, 1, 3, 0, 4, 3, 2, 4, 4, 4, 1, 1, 3, 0, 3, 4, 3, 2, 0, 3, 0, 3, 3, 4, 2, 0, 0, 1, 4, 2, 1, 0, 3, 3, 0, 1, 0, 2, 0, 2, 3, 3, 2, 0, 0, 1, 2, 1, 3, 3, 4, 3, 0, 2, 1, 0, 0, 0, 0, 4, 1, 1, 3, 2, 1, 2, 1, 3

Offset: 0

Jianing Song, Sep 07 2019

This square root of 6 in the 5-adic field ends with digit 4. The other, A324025, ends with digit 1.

			The solution to x^2 == 6 (mod 5^4) such that x == 4 (mod 5) is x == 109 (mod 5^4), and 109 is written as 414 in quinary, so the first four terms are 4, 1, 4 and 0.

Wikipedia, p-adic number

Cf. A324023, A324024.
Digits of 5-adic square roots:
A324029, A324030 (sqrt(-6));
A269591, A269592 (sqrt(-4));
A210850, A210851 (sqrt(-1));
A324025, this sequence (sqrt(6)).

a(n) = truncate(-sqrt(6+O(5^(n+1))))\5^n

a(n) = (A324024(n+1) - A324024(n))/5^n.
For n > 0, a(n) = 4 - A324025(n).
Equals A210850*A324029 = A210851*A324030, where each A-number represents a 5-adic number.

A325485 One of the four successive approximations up to 5^n for the 5-adic integer 6^(1/4). This is the 2 (mod 5) case (except for n = 0).

Original entry on oeis.org

0, 2, 22, 22, 397, 397, 6647, 6647, 319147, 319147, 6178522, 6178522, 103834772, 592116022, 3033522272, 9137037897, 70172194147, 222760084772, 3274517897272, 3274517897272, 60494976881647, 441964703444147, 1395639019850397, 3779824810866022, 51463540631178522

Offset: 0

Jianing Song, Sep 07 2019

nonn

For n > 0, a(n) is the unique number k in [1, 5^n] and congruent to 2 mod 5 such that k^4 - 6 is divisible by 5^n.

For k not divisible by 5, k is a fourth power in 5-adic field if and only if k == 1 (mod 5). If k is a fourth power in 5-adic field, then k has exactly 4 fourth-power roots.

			The unique number k in [1, 5^2] and congruent to 2 modulo 5 such that k^4 - 6 is divisible by 5^2 is k = 22, so a(2) = 22.
The unique number k in [1, 5^3] and congruent to 2 modulo 5 such that k^4 - 6 is divisible by 5^3 is also k = 22, so a(3) is also 22.

Wikipedia, p-adic number

Cf. A048898, A048899, A324024, A325489, A325490, A325491, A325492.
Approximations of p-adic fourth-power roots:
A325484, this sequence, A325486, A325487 (5-adic, 6^(1/4));
A324077, A324082, A324083, A324084 (13-adic, 3^(1/4)).

a(n) = lift(sqrtn(6+O(5^n), 4) * sqrt(-1+O(5^n)))

a(n) = A325484(n)*A048898(n) mod 13^n = A325485(n)*A048899(n) mod 13^n.
For n > 0, a(n) = 5^n - A325486(n).
a(n)^2 == A324024(n) (mod 5^n).

A324023 One of the two successive approximations up to 5^n for 5-adic integer sqrt(6). This is the 1 (mod 5) case (except for n = 0).

Original entry on oeis.org

0, 1, 16, 16, 516, 1766, 4891, 36141, 270516, 661141, 6520516, 35817391, 35817391, 768239266, 4430348641, 16637379891, 108190114266, 413365895516, 1939244801766, 9568639333016, 85862584645516, 371964879567391, 1802476354176766, 4186662145192391, 51870377965504891

Offset: 0

Jianing Song, Sep 07 2019

nonn

For n > 0, a(n) is the unique solution to x^2 == 6 (mod 5^n) in the range [0, 5^n - 1] and congruent to 1 modulo 5.

A324024 is the approximation (congruent to 4 mod 5) of another square root of 6 over the 5-adic field.

			16^2 = 256 = 10*5^2 + 6 = 2*5^3 + 6;
516^2 = 266256 = 426*5^4 + 6;
1766^2 = 3118756 = 998*5^5 + 6.

Wikipedia, p-adic number

Cf. A048898, A048899, A324025, A324026.
Approximations of 5-adic square roots:
A324027, A324028 (sqrt(-6));
A268922, A269590 (sqrt(-4));
A048898, A048899 (sqrt(-1));
this sequence, A324024 (sqrt(6)).

PARI
```
a(n) = truncate(sqrt(6+O(5^n)))
```

For n > 0, a(n) = 5^n - A324024(n).

a(n) = A048898(n)*A324028(n) mod 5^n = A048899(n)*A324027(n) mod 5^n.

A324025 Digits of one of the two 5-adic integers sqrt(6) that is related to A324023.

Original entry on oeis.org

1, 3, 0, 4, 2, 1, 2, 3, 1, 3, 3, 0, 3, 3, 2, 3, 2, 2, 2, 4, 3, 3, 1, 4, 0, 1, 2, 0, 0, 0, 3, 3, 1, 4, 1, 0, 1, 2, 4, 1, 4, 1, 1, 0, 2, 4, 4, 3, 0, 2, 3, 4, 1, 1, 4, 3, 4, 2, 4, 2, 1, 1, 2, 4, 4, 3, 2, 3, 1, 1, 0, 1, 4, 2, 3, 4, 4, 4, 4, 0, 3, 3, 1, 2, 3, 2, 3, 1

Offset: 0

Jianing Song, Sep 07 2019

This square root of 6 in the 5-adic field ends with digit 1. The other, A324026, ends with digit 4.

			The solution to x^2 == 6 (mod 5^4) such that x == 1 (mod 5) is x == 516 (mod 5^4), and 516 is written as 4031 in quinary, so the first four terms are 1, 3, 0 and 4.

Wikipedia, p-adic number

Cf. A324023, A324024.
Digits of 5-adic square roots:
A324029, A324030 (sqrt(-6));
A269591, A269592 (sqrt(-4));
A210850, A210851 (sqrt(-1));
this sequence, A324026 (sqrt(6)).

PARI
```
a(n) = truncate(sqrt(6+O(5^(n+1))))\5^n
```

a(n) = (A324023(n+1) - A324023(n))/5^n.
For n > 0, a(n) = 4 - A324026(n).
Equals A210850*A324030 = A210851*A324029, where each A-number represents a 5-adic number.

A325486 One of the four successive approximations up to 5^n for the 5-adic integer 6^(1/4). This is the 3 (mod 5) case (except for n = 0).

Original entry on oeis.org

0, 3, 3, 103, 228, 2728, 8978, 71478, 71478, 1633978, 3587103, 42649603, 140305853, 628587103, 3069993353, 21380540228, 82415696478, 540179368353, 540179368353, 15798968430853, 34872454758978, 34872454758978, 988546771165228, 8141104144212103, 8141104144212103

Offset: 0

Jianing Song, Sep 07 2019

nonn

For n > 0, a(n) is the unique number k in [1, 5^n] and congruent to 3 mod 5 such that k^4 - 6 is divisible by 5^n.

For k not divisible by 5, k is a fourth power in 5-adic field if and only if k == 1 (mod 5). If k is a fourth power in 5-adic field, then k has exactly 4 fourth-power roots.

			The unique number k in [1, 5^2] and congruent to 3 modulo 5 such that k^4 - 6 is divisible by 5^2 is k = 3, so a(2) = 3.
The unique number k in [1, 5^3] and congruent to 3 modulo 5 such that k^4 - 6 is divisible by 5^3 is k = 103, so a(3) = 103.

Wikipedia, p-adic number

Cf. A048898, A048899, A324024, A325489, A325490, A325491, A325492.
Approximations of p-adic fourth-power roots:
A325484, A325485, this sequence, A325487 (5-adic, 6^(1/4));
A324077, A324082, A324083, A324084 (13-adic, 3^(1/4)).

a(n) = lift(-sqrtn(6+O(5^n), 4) * sqrt(-1+O(5^n)))

a(n) = A325484(n)*A048899(n) mod 13^n = A325485(n)*A048898(n) mod 13^n.
For n > 0, a(n) = 5^n - A325485(n).
a(n)^2 == A324024(n) (mod 5^n).

A324027 One of the two successive approximations up to 5^n for 5-adic integer sqrt(-6). This is the 2 (mod 5) case (except for n = 0).

Original entry on oeis.org

0, 2, 12, 37, 162, 1412, 10787, 42037, 354537, 1526412, 3479537, 3479537, 3479537, 247620162, 3909729537, 10013245162, 101565979537, 711917542037, 2237796448287, 13681888245162, 51828860901412, 337931155823287, 1291605472229537, 10828348636292037, 58512064456604537

Offset: 0

Jianing Song, Sep 07 2019

nonn

For n > 0, a(n) is the unique solution to x^2 == -6 (mod 5^n) in the range [0, 5^n - 1] and congruent to 2 modulo 5.

A324028 is the approximation (congruent to 3 mod 5) of another square root of 6 over the 5-adic field.

			12^2 = 144 = 6*5^2 - 6;
37^2 = 1369 = 11*5^3 - 6;
162^2 = 26244 = 42*5^4 - 6.

Wikipedia, p-adic number

Cf. A048898, A048899, A324029, A324030.
Approximations of 5-adic square roots:
this sequence, A324028 (sqrt(-6));
A268922, A269590 (sqrt(-4));
A048898, A048899 (sqrt(-1));
A324023, A324024 (sqrt(6)).

PARI
```
a(n) = truncate(sqrt(-6+O(5^n)))
```

For n > 0, a(n) = 5^n - A324028(n).

a(n) = A048898(n)*A324023(n) mod 5^n = A048899(n)*A324024(n) mod 5^n.

A324028 One of the two successive approximations up to 5^n for 5-adic integer sqrt(-6). This is the 3 (mod 5) case (except for n = 0).

Original entry on oeis.org

0, 3, 13, 88, 463, 1713, 4838, 36088, 36088, 426713, 6286088, 45348588, 240661088, 973082963, 2193786088, 20504332963, 51021911088, 51021911088, 1576900817338, 5391598082963, 43538570739213, 138906002379838, 1092580318786088, 1092580318786088, 1092580318786088

Offset: 0

Jianing Song, Sep 07 2019

nonn

For n > 0, a(n) is the unique solution to x^2 == -6 (mod 5^n) in the range [0, 5^n - 1] and congruent to 3 modulo 5.

A324027 is the approximation (congruent to 3 mod 5) of another square root of -6 over the 5-adic field.

			13^2 = 169 = 7*5^2 - 6;
88^2 = 7744 = 62*5^3 - 6;
463^2 = 214369 = 343*5^4 - 6.

Wikipedia, p-adic number

Cf. A048898, A048899, A324029, A324030.
Approximations of 5-adic square roots:
A324027, sequence (sqrt(-6));
A268922, A269590 (sqrt(-4));
A048898, A048899 (sqrt(-1));
A324023, A324024 (sqrt(6)).

PARI
```
a(n) = truncate(-sqrt(-6+O(5^n)))
```

For n > 0, a(n) = 5^n - A324027(n).

a(n) = A048898(n)*A324024(n) mod 5^n = A048899(n)*A324023(n) mod 5^n.

A327303 One of the two successive approximations up to 5^n for the 5-adic integer sqrt(-9). This is the 4 (mod 5) case (except for n = 0).

Original entry on oeis.org

0, 4, 4, 79, 79, 79, 3204, 18829, 331329, 1112579, 1112579, 20643829, 118300079, 850721954, 3292128204, 27706190704, 149776503204, 302364393829, 1065303846954, 8694698378204, 46841671034454, 332943965956329, 332943965956329, 5101315547987579, 28943173458143829

Offset: 0

Jianing Song, Sep 16 2019

nonn

a(n) is the unique number k in [1, 5^n] and congruent to 4 mod 5 such that k^2 + 9 is divisible by 5^n.

			The unique number k in {4, 9, 14, 19, 24} such that k^2 + 9 is divisible by 25 is k = 4, so a(2) = 4.
The unique number k in {4, 29, 54, 79, 104} such that k^2 + 9 is divisible by 125 is k = 79, so a(3) = 46.
The unique number k in {79, 204, 329, 454, 579} such that k^2 + 9 is divisible by 625 is k = 79, so a(4) = 79.

Robert Israel, Table of n, a(n) for n = 0..1424
G. P. Michon, Introduction to p-adic integers, Numericana.

For the digits of sqrt(-9) see A327304 and A327305.
Approximations of 5-adic square roots:
A327302, this sequence (sqrt(-9));
A324027, A324028 (sqrt(-6));
A268922, A269590 (sqrt(-4));
A048898, A048899 (sqrt(-1));
A324023, A324024 (sqrt(6)).

R:= [padic:-rootp(x^2+9,5,101)]:
R:= op(select(t -> padic:-ratvaluep(t,1)=4, R)):
seq(padic:-ratvaluep(R,n),n=0..100); # Robert Israel, Jan 16 2023

PARI
```
a(n) = truncate(-sqrt(-9+O(5^n)))
```

a(1) = 4; for n >= 2, a(n) is the unique number k in {a(n-1) + m*5^(n-1) : m = 0, 1, 2, 3, 4} such that k^2 + 9 is divisible by 5^n.

For n > 0, a(n) = 5^n - A327302(n).

A327302 One of the two successive approximations up to 5^n for the 5-adic integer sqrt(-9). This is the 1 (mod 5) case (except for n = 0).

Original entry on oeis.org

0, 1, 21, 46, 546, 3046, 12421, 59296, 59296, 840546, 8653046, 28184296, 125840546, 369981171, 2811387421, 2811387421, 2811387421, 460575059296, 2749393418671, 10378787949921, 48525760606171, 143893192246796, 2051241825059296, 6819613407090546, 30661471317246796

Offset: 0

Jianing Song, Sep 16 2019

nonn

a(n) is the unique number k in [1, 5^n] and congruent to 1 mod 5 such that k^2 + 9 is divisible by 5^n.

			The unique number k in {1, 6, 11, 16, 21} such that k^2 + 9 is divisible by 25 is k = 21, so a(2) = 21.
The unique number k in {21, 46, 71, 96, 121} such that k^2 + 9 is divisible by 125 is k = 46, so a(3) = 46.
The unique number k in {46, 171, 296, 421, 546} such that k^2 + 9 is divisible by 625 is k = 546, so a(4) = 546.

G. P. Michon, Introduction to p-adic integers, Numericana.

For the digits of sqrt(-9) see A327304 and A327305.
Approximations of 5-adic square roots:
this sequence, A327303 (sqrt(-9));
A324027, A324028 (sqrt(-6));
A268922, A269590 (sqrt(-4));
A048898, A048899 (sqrt(-1));
A324023, A324024 (sqrt(6)).

PARI
```
a(n) = truncate(sqrt(-9+O(5^n)))
```

a(1) = 1; for n >= 2, a(n) is the unique number k in {a(n-1) + m*5^(n-1) : m = 0, 1, 2, 3, 4} such that k^2 + 9 is divisible by 5^n.

For n > 0, a(n) = 5^n - A327303(n).

cp's OEIS Frontend

A324026 Digits of one of the two 5-adic integers sqrt(6) that is related to A324024.

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A325485 One of the four successive approximations up to 5^n for the 5-adic integer 6^(1/4). This is the 2 (mod 5) case (except for n = 0).

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A324023 One of the two successive approximations up to 5^n for 5-adic integer sqrt(6). This is the 1 (mod 5) case (except for n = 0).

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A324025 Digits of one of the two 5-adic integers sqrt(6) that is related to A324023.

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A325486 One of the four successive approximations up to 5^n for the 5-adic integer 6^(1/4). This is the 3 (mod 5) case (except for n = 0).

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A324027 One of the two successive approximations up to 5^n for 5-adic integer sqrt(-6). This is the 2 (mod 5) case (except for n = 0).

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A324028 One of the two successive approximations up to 5^n for 5-adic integer sqrt(-6). This is the 3 (mod 5) case (except for n = 0).

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A327303 One of the two successive approximations up to 5^n for the 5-adic integer sqrt(-9). This is the 4 (mod 5) case (except for n = 0).

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