I'm working on modernizing Rosetta Code's infrastructure. Starting with communications. Please accept this time-limited open invite to RC's Slack.. --Michael Mol (talk) 20:59, 30 May 2020 (UTC)

Pell numbers

From Rosetta Code
Task
Pell numbers
You are encouraged to solve this task according to the task description, using any language you may know.

Pell numbers are an infinite sequence of integers that comprise the denominators of the closest rational approximations to the square root of 2 but have many other interesting uses and relationships.

The numerators of each term of rational approximations to the square root of 2 may also be derived from Pell numbers, or may be found by taking half of each term of the related sequence: Pell-Lucas or Pell-companion numbers.


The Pell numbers: 0, 1, 2, 5, 12, 29, 70, etc., are defined by the recurrence relation:


P0 = 0;
P1 = 1;
Pn = 2 × Pn-1 + Pn-2;

Or, may also be expressed by the closed form formula:


Pn = ((1 + √2)n - (1 - √2)n) / (2 × √2);


Pell-Lucas or Pell-companion numbers: 2, 2, 6, 14, 34, 82, etc., are defined by a very similar recurrence relation, differing only in the first two terms:


Q0 = 2;
Q1 = 2;
Qn = 2 × Qn-1 + Qn-2;

Or, may also be expressed by the closed form formula:


Qn = (1 + √2)n + (1 - √2)n;

or


Qn = P2n / Pn;


The sequence of rational approximations to the square root of 2 begins:


1/1, 3/2, 7/5, 17/12, 41/29, ...

Starting from n = 1, for each term, the denominator is Pn and the numerator is Qn / 2 or Pn-1 + Pn.



Pell primes are Pell numbers that are prime. Pell prime indices are the indices of the primes in the Pell numbers sequence. Every Pell prime index is prime, though not every prime index corresponds to a prime Pell number.


If you take the sum S of the first 4n + 1 Pell numbers, the sum of the terms P2n and P2n + 1 will form the square root of S.

For instance, the sum of the Pell numbers up to P5; 0 + 1 + 2 + 5 + 12 + 29 == 49, is the square of P2 + P3 == 2 + 5 == 7. The sequence of numbers formed by the sums P2n + P2n + 1 are known as Newman-Shank-Williams numbers or NSW numbers.


Pell numbers may also be used to find Pythagorean triple near isosceles right triangles; right triangles whose legs differ by exactly 1. E.G.: (3,4,5), (20,21,29), (119,120,169), etc.

For n > 0, each right triangle hypotenuse is P2n + 1. The shorter leg length is the sum of the terms up to P2n + 1. The longer leg length is 1 more than that.


Task
  • Find and show at least the first 10 Pell numbers.
  • Find and show at least the first 10 Pell-Lucas numbers.
  • Use the Pell (and optionally, Pell-Lucas) numbers sequence to find and show at least the first 10 rational approximations to √2 in both rational and decimal representation.
  • Find and show at least the first 10 Pell primes.
  • Find and show at least the first 10 indices of Pell primes.
  • Find and show at least the first 10 Newman-Shank-Williams numbers
  • Find and show at least the first 10 Pythagorean triples corresponding to near isosceles right triangles.


See also



FreeBASIC[edit]

Translation of: Phix
#define isOdd(a) (((a) and 1) <> 0)
 
Function isPrime(Byval ValorEval As Integer) As Boolean
If ValorEval < 2 Then Return False
If ValorEval Mod 2 = 0 Then Return ValorEval = 2
If ValorEval Mod 3 = 0 Then Return ValorEval = 3
Dim d As Integer = 5
While d * d <= ValorEval
If ValorEval Mod d = 0 Then Return False Else d += 2
If ValorEval Mod d = 0 Then Return False Else d += 4
Wend
Return True
End Function
 
Dim As Integer n
Dim As Integer p(0 To 40), pl(0 To 40)
p(0)= 0: p(1) = 1
pl(0) = 2: pl(1) = 2
For n = 2 To 40
p(n) = 2 * p(n-1) + p(n-2)
pl(n) = 2 * pl(n-1) + pl(n-2)
Next n
 
Print "First 20 Pell numbers: "
For n = 0 To 19 : Print p(n); : Next n
Print !"\n\nFirst 20 Pell-Lucas: "
For n = 0 To 19 : Print pl(n); : Next n
 
Print !"\n\nFirst 20 rational approximations of sqrt(2) (" & Str(Sqr(2)) & "): "
For n = 1 To 20
Dim As Integer j = pl(n)/2, d = p(n)
Print Using " &/& ~= &"; j; d; j/d
Next n
 
Print !"\nFirst 6 Pell primes: [for the limitations of the FB standard library]"
Dim as Integer pdx = 2
Dim As Byte c = 0
Dim As Ulongint ppdx(1 to 20)
do
If isPrime(p(pdx)) Then
If isPrime(pdx) Then ppdx(c) = pdx : End If
Print p(pdx)
c += 1
End If
pdx += 1
loop until c = 6
 
Print !"\nIndices of first 6 Pell primes: [for the limitations of the FB standard library]"
For n = 0 To 5 : Print " "; ppdx(n); : Next n
 
Dim As Ulongint nsw(0 To 20)
For n = 0 To 19
nsw(n) = p(2*n) + p(2*n+1)
Next n
Print !"\n\nFirst 20 Newman-Shank-Williams numbers: "
For n = 0 To 19 : Print " "; nsw(n); : Next n
 
Print !"\n\nFirst 20 near isosceles right triangles:"
Dim As Integer i0 = 0, i1 = 1, i2, t = 1, i = 2, found = 0
Do While found < 20
i2 = i1*2 + i0
If isOdd(i) Then
Print Using " [&, &, &]"; t; t+1 ; i2
found += 1
End If
t += i2
i0 = i1 : i1 = i2
i += 1
Loop
Sleep
Output:
First 20 Pell numbers:
 0 1 2 5 12 29 70 169 408 985 2378 5741 13860 33461 80782 195025 470832 1136689 2744210 6625109

First 20 Pell-Lucas:
 2 2 6 14 34 82 198 478 1154 2786 6726 16238 39202 94642 228486 551614 1331714 3215042 7761798 18738638

First 20 rational approximations of sqrt(2) (1.414213562373095):
 1/1 ~= 1
 3/2 ~= 1.5
 7/5 ~= 1.4
 17/12 ~= 1.416666666666667
 41/29 ~= 1.413793103448276
 99/70 ~= 1.414285714285714
 239/169 ~= 1.414201183431953
 577/408 ~= 1.41421568627451
 1393/985 ~= 1.414213197969543
 3363/2378 ~= 1.41421362489487
 8119/5741 ~= 1.414213551646055
 19601/13860 ~= 1.414213564213564
 47321/33461 ~= 1.41421356205732
 114243/80782 ~= 1.414213562427273
 275807/195025 ~= 1.414213562363799
 665857/470832 ~= 1.41421356237469
 1607521/1136689 ~= 1.414213562372821
 3880899/2744210 ~= 1.414213562373142
 9369319/6625109 ~= 1.414213562373087
 22619537/15994428 ~= 1.414213562373096

First 6 Pell primes: [for the limitations of the FB standard library]
 2
 5
 29
 5741
 33461
 44560482149

Indices of first 6 Pell primes: [for the limitations of the FB standard library]
 2 3 5 11 13 29

First 20 Newman-Shank-Williams numbers:
 1 7 41 239 1393 8119 47321 275807 1607521 9369319 54608393 318281039 1855077841 10812186007 63018038201 367296043199 2140758220993 12477253282759 72722761475561 423859315570607

First 20 near isosceles right triangles:
 [3, 4, 5]
 [20, 21, 29]
 [119, 120, 169]
 [696, 697, 985]
 [4059, 4060, 5741]
 [23660, 23661, 33461]
 [137903, 137904, 195025]
 [803760, 803761, 1136689]
 [4684659, 4684660, 6625109]
 [27304196, 27304197, 38613965]
 [159140519, 159140520, 225058681]
 [927538920, 927538921, 1311738121]
 [5406093003, 5406093004, 7645370045]
 [31509019100, 31509019101, 44560482149]
 [183648021599, 183648021600, 259717522849]
 [1070379110496, 1070379110497, 1513744654945]
 [6238626641379, 6238626641380, 8822750406821]
 [36361380737780, 36361380737781, 51422757785981]
 [211929657785303, 211929657785304, 299713796309065]
 [1235216565974040, 1235216565974041, 1746860020068409]


Go[edit]

Translation of: Wren
Library: Go-rcu
package main
 
import (
"fmt"
"math/big"
"rcu"
)
 
func main() {
p := make([]int64, 40)
p[1] = 1
for i := 2; i < 40; i++ {
p[i] = 2*p[i-1] + p[i-2]
}
fmt.Println("The first 20 Pell numbers are:")
fmt.Println(p[0:20])
 
q := make([]int64, 40)
q[0] = 2
q[1] = 2
for i := 2; i < 40; i++ {
q[i] = 2*q[i-1] + q[i-2]
}
fmt.Println("\nThe first 20 Pell-Lucas numbers are:")
fmt.Println(q[0:20])
 
fmt.Println("\nThe first 20 rational approximations of √2 (1.4142135623730951) are:")
for i := 1; i <= 20; i++ {
r := big.NewRat(q[i]/2, p[i])
fmt.Printf("%-17s ≈ %-18s\n", r, r.FloatString(16))
}
 
fmt.Println("\nThe first 15 Pell primes are:")
p0 := big.NewInt(0)
p1 := big.NewInt(1)
p2 := big.NewInt(0)
two := big.NewInt(2)
indices := make([]int, 15)
for index, count := 2, 0; count < 15; index++ {
p2.Mul(p1, two)
p2.Add(p2, p0)
if rcu.IsPrime(index) && p2.ProbablyPrime(15) {
fmt.Println(p2)
indices[count] = index
count++
}
p0.Set(p1)
p1.Set(p2)
}
 
fmt.Println("\nIndices of the first 15 Pell primes are:")
fmt.Println(indices)
 
fmt.Println("\nFirst 20 Newman-Shank_Williams numbers:")
nsw := make([]int64, 20)
for n := 0; n < 20; n++ {
nsw[n] = p[2*n] + p[2*n+1]
}
fmt.Println(nsw)
 
fmt.Println("\nFirst 20 near isosceles right triangles:")
u0 := 0
u1 := 1
sum := 1
for i := 2; i < 43; i++ {
u2 := u1*2 + u0
if i%2 == 1 {
fmt.Printf("(%d, %d, %d)\n", sum, sum+1, u2)
}
sum += u2
u0 = u1
u1 = u2
}
}
Output:
The first 20 Pell numbers are:
[0 1 2 5 12 29 70 169 408 985 2378 5741 13860 33461 80782 195025 470832 1136689 2744210 6625109]

The first 20 Pell-Lucas numbers are:
[2 2 6 14 34 82 198 478 1154 2786 6726 16238 39202 94642 228486 551614 1331714 3215042 7761798 18738638]

The first 20 rational approximations of √2 (1.4142135623730951) are:
1/1               ≈ 1.0000000000000000
3/2               ≈ 1.5000000000000000
7/5               ≈ 1.4000000000000000
17/12             ≈ 1.4166666666666667
41/29             ≈ 1.4137931034482759
99/70             ≈ 1.4142857142857143
239/169           ≈ 1.4142011834319527
577/408           ≈ 1.4142156862745098
1393/985          ≈ 1.4142131979695431
3363/2378         ≈ 1.4142136248948696
8119/5741         ≈ 1.4142135516460547
19601/13860       ≈ 1.4142135642135642
47321/33461       ≈ 1.4142135620573205
114243/80782      ≈ 1.4142135624272734
275807/195025     ≈ 1.4142135623637995
665857/470832     ≈ 1.4142135623746899
1607521/1136689   ≈ 1.4142135623728214
3880899/2744210   ≈ 1.4142135623731420
9369319/6625109   ≈ 1.4142135623730870
22619537/15994428 ≈ 1.4142135623730964

The first 15 Pell primes are:
2
5
29
5741
33461
44560482149
1746860020068409
68480406462161287469
13558774610046711780701
4125636888562548868221559797461449
4760981394323203445293052612223893281
161733217200188571081311986634082331709
2964793555272799671946653940160950323792169332712780937764687561
677413820257085084326543915514677342490435733542987756429585398537901
4556285254333448771505063529048046595645004014152457191808671945330235841

Indices of the first 15 Pell primes are:
[2 3 5 11 13 29 41 53 59 89 97 101 167 181 191]

First 20 Newman-Shank_Williams numbers:
[1 7 41 239 1393 8119 47321 275807 1607521 9369319 54608393 318281039 1855077841 10812186007 63018038201 367296043199 2140758220993 12477253282759 72722761475561 423859315570607]

First 20 near isosceles right triangles:
(3, 4, 5)
(20, 21, 29)
(119, 120, 169)
(696, 697, 985)
(4059, 4060, 5741)
(23660, 23661, 33461)
(137903, 137904, 195025)
(803760, 803761, 1136689)
(4684659, 4684660, 6625109)
(27304196, 27304197, 38613965)
(159140519, 159140520, 225058681)
(927538920, 927538921, 1311738121)
(5406093003, 5406093004, 7645370045)
(31509019100, 31509019101, 44560482149)
(183648021599, 183648021600, 259717522849)
(1070379110496, 1070379110497, 1513744654945)
(6238626641379, 6238626641380, 8822750406821)
(36361380737780, 36361380737781, 51422757785981)
(211929657785303, 211929657785304, 299713796309065)
(1235216565974040, 1235216565974041, 1746860020068409)

J[edit]

As detailed in the task description, there's a variety of ways to compute these values.

For example:

nextPell=: , 1 2+/ .*_2&{. NB. pell, list extender
Pn=: (%:8) %~(1+%:2)&^ - (1-%:2)&^ NB. pell, closed form
Qn=: (1+%:2)&^ + (1-%:2)&^ NB. pell lucas, closed form
QN=: +: %&Pn ] NB. pell lucas, closed form
qn=: 2 * (+&Pn <:) NB. pell lucas, closed form

Thus:

   nextPell^:9(0 1)
0 1 2 5 12 29 70 169 408 985 2378
Pn i.11
0 1 2 5 12 29 70 169 408 985 2378
nextPell^:9(2 2)
2 2 6 14 34 82 198 478 1154 2786 6726
Qn i.11
2 2 6 14 34 82 198 478 1154 2786 6726
QN i.11
0 2 6 14 34 82 198 478 1154 2786 6726
qn i.11
2 2 6 14 34 82 198 478 1154 2786 6726

QN (which is defined as P2n/Pn) doesn't get the first element of the pell lucas sequence right. We could fix this by changing the definition:

QN=: 2 >. +: %&Pn ]
QN i.11
2 2 6 14 34 82 198 478 1154 2786 6726

Continuing... the first ten rational approximations to √2 here would be:

   }.(%~ _1}. +//[email protected],:~) nextPell^:9(0 1)
1 1.5 1.4 1.41667 1.41379 1.41429 1.4142 1.41422 1.41421 1.41421
}.(%~ _1}. +//[email protected],:~) nextPell^:9(0 1x)
1 3r2 7r5 17r12 41r29 99r70 239r169 577r408 1393r985 3363r2378

The first ten pell primes are:

   10{.(#~ 1&p:)nextPell^:99(0 1x)
2 5 29 5741 33461 44560482149 1746860020068409 68480406462161287469 13558774610046711780701 4125636888562548868221559797461449

Their indices are:

   10{.I. 1&p:nextPell^:99(0 1x)
2 3 5 11 13 29 41 53 59 89

The NSW numbers are the sums of (non-overlapping) pairs of pell numbers, or:

   _2 +/\ nextPell^:20(0 1x)
1 7 41 239 1393 8119 47321 275807 1607521 9369319 54608393
 

The first ten pell based pythogorean triples would be:

   }.(21$1 0)#|:(}.,~0 1+/+/\@}:)nextPell^:(20)0 1
3 4 5
20 21 29
119 120 169
696 697 985
4059 4060 5741
23660 23661 33461
137903 137904 195025
803760 803761 1136689
4684659 4684660 6625109
27304196 27304197 38613965

Julia[edit]

using Primes
 
function pellnumbers(wanted)
pells = [0, 1]
wanted < 3 && return pells[1:wanted]
while length(pells) < wanted
push!(pells, 2 * pells[end] + pells[end - 1])
end
return pells
end
 
function pelllucasnumbers(wanted)
pelllucas = [2, 2]
wanted < 3 && return pelllucas[1:wanted]
while length(pelllucas) < wanted
push!(pelllucas, 2 * pelllucas[end] + pelllucas[end - 1])
end
return pelllucas
end
 
function pellprimes(wanted)
i, lastpell, lastlastpell, primeindices, pellprimes = 1, big"1", big"0", Int[], BigInt[]
while length(primeindices) < wanted
pell = 2 * lastpell + lastlastpell
i += 1
if isprime(pell)
push!(primeindices, i)
push!(pellprimes, pell)
end
lastpell, lastlastpell = pell, lastpell
end
return primeindices, pellprimes
end
 
function approximationsqrt2(wanted)
p, q = pellnumbers(wanted + 1), pelllucasnumbers(wanted + 1)
return map(r -> "$r ≈ $(Float64(r))", [(q[n] // 2) // p[n] for n in 2:wanted+1])
end
 
function newmanshankwilliams(wanted)
pells = pellnumbers(wanted * 2 + 1)
return [pells[2i - 1] + pells[2i] for i in 1:wanted]
end
 
function nearisosceles(wanted)
pells = pellnumbers((wanted + 1) * 2 + 1)
return map(x -> (last(x), last(x) + 1, first(x)),
[(pells[2i], sum(pells[1:2i-1])) for i in 2:wanted+1])
end
 
function printrows(title, vec, columnsize = 8, columns = 10, rjust=false)
println(title)
for (i, n) in enumerate(vec)
print((rjust ? lpad : rpad)(n, columnsize), i % columns == 0 ? "\n" : "")
end
println()
end
 
printrows("Twenty Pell numbers:", pellnumbers(20))
printrows("Twenty Pell-Lucas numbers:", pelllucasnumbers(20))
printrows("Twenty approximations of √2:", approximationsqrt2(20), 44, 2)
pindices, pprimes = pellprimes(15)
printrows("Fifteen Pell primes:", pprimes, 90, 1)
printrows("Fifteen Pell prime zero-based indices:", pindices, 4, 15)
printrows("Twenty Newman-Shank-Williams numbers:", newmanshankwilliams(20), 17, 5)
printrows("Twenty near isosceles triangle triplets:", nearisosceles(20), 52, 2)
 
Output:
Twenty Pell numbers:
0       1       2       5       12      29      70      169     408     985     
2378    5741    13860   33461   80782   195025  470832  1136689 2744210 6625109 

Twenty Pell-Lucas numbers:
2       2       6       14      34      82      198     478     1154    2786    
6726    16238   39202   94642   228486  551614  1331714 3215042 7761798 18738638

Twenty approximations of √2:
1//1 ≈ 1.0                                  3//2 ≈ 1.5
7//5 ≈ 1.4                                  17//12 ≈ 1.4166666666666667
41//29 ≈ 1.4137931034482758                 99//70 ≈ 1.4142857142857144
239//169 ≈ 1.4142011834319526               577//408 ≈ 1.4142156862745099
1393//985 ≈ 1.4142131979695431              3363//2378 ≈ 1.4142136248948696
8119//5741 ≈ 1.4142135516460548             19601//13860 ≈ 1.4142135642135643
47321//33461 ≈ 1.4142135620573204           114243//80782 ≈ 1.4142135624272734
275807//195025 ≈ 1.4142135623637995         665857//470832 ≈ 1.4142135623746899
1607521//1136689 ≈ 1.4142135623728214       3880899//2744210 ≈ 1.414213562373142
9369319//6625109 ≈ 1.414213562373087        22619537//15994428 ≈ 1.4142135623730965

Fifteen Pell primes:
2
5
29
5741
33461
44560482149
1746860020068409
68480406462161287469
13558774610046711780701
4125636888562548868221559797461449
4760981394323203445293052612223893281
161733217200188571081311986634082331709
2964793555272799671946653940160950323792169332712780937764687561
677413820257085084326543915514677342490435733542987756429585398537901
4556285254333448771505063529048046595645004014152457191808671945330235841

Fifteen Pell prime zero-based indices:
2   3   5   11  13  29  41  53  59  89  97  101 167 181 191

Twenty Newman-Shank-Williams numbers:
1                7                41               239              1393
8119             47321            275807           1607521          9369319
54608393         318281039        1855077841       10812186007      63018038201
367296043199     2140758220993    12477253282759   72722761475561   423859315570607

Twenty near isosceles triangle triplets:
(3, 4, 5)                                           (20, 21, 29)
(119, 120, 169)                                     (696, 697, 985)
(4059, 4060, 5741)                                  (23660, 23661, 33461)
(137903, 137904, 195025)                            (803760, 803761, 1136689)
(4684659, 4684660, 6625109)                         (27304196, 27304197, 38613965)
(159140519, 159140520, 225058681)                   (927538920, 927538921, 1311738121)
(5406093003, 5406093004, 7645370045)                (31509019100, 31509019101, 44560482149)
(183648021599, 183648021600, 259717522849)          (1070379110496, 1070379110497, 1513744654945)
(6238626641379, 6238626641380, 8822750406821)       (36361380737780, 36361380737781, 51422757785981)
(211929657785303, 211929657785304, 299713796309065) (1235216565974040, 1235216565974041, 1746860020068409)

Perl[edit]

Library: ntheory
use strict;
use warnings;
use feature <state say>;
use bignum;
use ntheory 'is_prime';
use List::Util <sum head>;
use List::Lazy 'lazy_list';
 
my $upto = 20;
 
my $pell_recur = lazy_list { state @p = (0, 1); push @p, 2*$p[1] + $p[0]; shift @p };
my $pell_lucas_recur = lazy_list { state @p = (2, 2); push @p, 2*$p[1] + $p[0]; shift @p };
 
my @pell;
push @pell, $pell_recur->next() for 1..1500; #wart;
 
my @pell_lucas;
push @pell_lucas, $pell_lucas_recur->next() for 1..$upto+1;
 
say "First $upto Pell numbers:";
say join ' ', @pell[0..$upto-1];
 
say "\nFirst $upto Pell-Lucas numbers:";
say join ' ', @pell_lucas[0..$upto-1];
 
say "\nFirst $upto rational approximations of √2:";
say sprintf "%d/%d - %1.16f", $pell[$_-1] + $pell[$_], $pell[$_], ($pell[$_-1]+$pell[$_])/$pell[$_] for 1..$upto;
 
say "\nFirst $upto Pell primes:";
say join "\n", head $upto, grep { is_prime $_ } @pell;
 
say "\nIndices of first $upto Pell primes:";
say join ' ', head $upto, grep { is_prime($pell[$_]) and $_ } 0..$#pell;
 
say "\nFirst $upto Newman-Shank-Williams numbers:";
say join ' ', map { $pell[2 * $_] + $pell[2 * $_+1] } 0..$upto-1;
 
say "\nFirst $upto near isoceles right tringles:";
map {
my $y = 2*$_ + 1;
my $x = sum @pell[0..$y-1];
printf "(%d, %d, %d)\n", $x, $x+1, $pell[$y]
} 1..$upto;
Output:
First 20 Pell numbers:
0 1 2 5 12 29 70 169 408 985 2378 5741 13860 33461 80782 195025 470832 1136689 2744210 6625109

First 20 Pell-Lucas numbers:
2 2 6 14 34 82 198 478 1154 2786 6726 16238 39202 94642 228486 551614 1331714 3215042 7761798 18738638

First 20 rational approximations of √2:
1/1 - 1.0000000000000000
3/2 - 1.5000000000000000
7/5 - 1.4000000000000000
17/12 - 1.4166666666666667
41/29 - 1.4137931034482758
99/70 - 1.4142857142857144
239/169 - 1.4142011834319526
577/408 - 1.4142156862745099
1393/985 - 1.4142131979695431
3363/2378 - 1.4142136248948696
8119/5741 - 1.4142135516460548
19601/13860 - 1.4142135642135643
47321/33461 - 1.4142135620573204
114243/80782 - 1.4142135624272734
275807/195025 - 1.4142135623637995
665857/470832 - 1.4142135623746899
1607521/1136689 - 1.4142135623728214
3880899/2744210 - 1.4142135623731420
9369319/6625109 - 1.4142135623730870
22619537/15994428 - 1.4142135623730965

First 20 Pell primes:
2
5
29
5741
33461
44560482149
1746860020068409
68480406462161287469
13558774610046711780701
4125636888562548868221559797461449
4760981394323203445293052612223893281
161733217200188571081311986634082331709
2964793555272799671946653940160950323792169332712780937764687561
677413820257085084326543915514677342490435733542987756429585398537901
4556285254333448771505063529048046595645004014152457191808671945330235841
54971607658948646301386783144964782698772613513307493180078896702918825851648683235325858118170150873214978343601463118106546653220435805362395962991295556488036606954237309847762149971207793263738989
14030291214037674827921599320400561033992948898216351802670122530401263880575255235196727095109669287799074570417579539629351231775861429098849146880746524269269235328805333087546933690012894630670427794266440579064751300508834822795162874147983974059159392260220762973563561382652223360667198516093199367134903695783143116067743023134509886357032327271649
2434804314652199381956027075145741187716221548707931096877274520825143228915116227412484991366386864484767844200542482630246332092069382947111767723898168035847078557798454111405556629400142434835890123610082763986456199467423944182141028870863302603437534363208996458153115358483747994095302552907353919742211197822912892578751357668345638404394626711701120567186348490247426710813709165801137112237291901437566040249805155494297005186344325519103590369653438042689
346434895614929444828445967916634653215454504812454865104089892164276080684080254746939261017687341632569935171059945916359539268094914543114024020158787741692287531903178502306292484033576487391159597130834863729261484555671037916432206867189514675750227327687799973497042239286045783392065227614939379139866240959756584073664244580698830046194724340448293320938108876004367449471918175071251610962540447986139876845105399212429593945098472125140242905536711601925585608153109062121115635939797709
32074710952523740376423283403256578238321646122759160107427497117576305397686814013623874765833543023397971470911301264845142006214276865917420065183527313421909784286074786922242104480428021290764613639424408361555091057197776876849282654018358993099016644054242247557103410808928387071991436781136646322261169941417916607548507224950058710729258466238995253184617782314756913932650536663800753256087990078866003788647079369825102832504351225446531057648755795494571534144773842019836572551455718577614678081652481281009

Indices of first 20 Pell primes:
2 3 5 11 13 29 41 53 59 89 97 101 167 181 191 523 929 1217 1301 1361

First 20 Newman-Shank-Williams numbers:
1 7 41 239 1393 8119 47321 275807 1607521 9369319 54608393 318281039 1855077841 10812186007 63018038201 367296043199 2140758220993 12477253282759 72722761475561 423859315570607

First 20 near isosceles right triangles:
(3, 4, 5)
(20, 21, 29)
(119, 120, 169)
(696, 697, 985)
(4059, 4060, 5741)
(23660, 23661, 33461)
(137903, 137904, 195025)
(803760, 803761, 1136689)
(4684659, 4684660, 6625109)
(27304196, 27304197, 38613965)
(159140519, 159140520, 225058681)
(927538920, 927538921, 1311738121)
(5406093003, 5406093004, 7645370045)
(31509019100, 31509019101, 44560482149)
(183648021599, 183648021600, 259717522849)
(1070379110496, 1070379110497, 1513744654945)
(6238626641379, 6238626641380, 8822750406821)
(36361380737780, 36361380737781, 51422757785981)
(211929657785303, 211929657785304, 299713796309065)
(1235216565974040, 1235216565974041, 1746860020068409)

Phix[edit]

with javascript_semantics
sequence p = {0,1},
         pl = {2,2}
for i=2 to 41 do
    p &= 2*p[i]+p[i-1]
    pl &= 2*pl[i]+pl[i-1]
end for
printf(1,"First 20 Pell numbers: %s\n",{join_by(p[1..20],1,20," ",fmt:="%d")})
printf(1,"First 20 Pell-Lucas: %s\n",{join_by(pl[1..20],1,20," ",fmt:="%d")})
printf(1,"First 20 rational approximations of sqrt(2) (%.16f):\n",{sqrt(2)})
for i=2 to 21 do
    integer n = pl[i]/2, d = p[i]
    printf(1,"%d/%d ~= %.16g\n", {n,d,n/d})
end for
printf(1,"\nFirst 20 Pell primes:\n")
include mpfr.e
mpz {p0,p1,p2} = mpz_inits(3,{0,1,0})
sequence ppdx = {}
integer pdx = 2
while length(ppdx)<20 do
    mpz_mul_si(p2,p1,2)
    mpz_add(p2,p2,p0)
    if is_prime(pdx) and mpz_prime(p2) then
        printf(1,"%s\n",mpz_get_short_str(p2))
        ppdx = append(ppdx,sprintf("%d",pdx))
    end if
    pdx += 1
    mpz_set(p0,p1)
    mpz_set(p1,p2)
end while
printf(1,"\nIndices of first 20 Pell primes: %s\n",join(ppdx," "))
sequence nsw = {}
for n=1 to 20 do nsw = append(nsw,sprintf("%d",p[2*n]+p[2*n-1])) end for
nsw[8..-3] = {"..."}
printf(1,"\nFirst 20 Newman-Shank-Williams numbers: %s\n",{join(nsw," ")})
printf(1,"\nFirst 20 near isosceles right triangles:\n")
for i=4 to 42 by 2 do
    atom side = sum(p[1..i-1]), hypot = p[i]
    printf(1,"[%d, %d, %d]\n", {side,side+1,hypot})
end for
Output:
First 20 Pell numbers: 0 1 2 5 12 29 70 169 408 985 2378 5741 13860 33461 80782 195025 470832 1136689 2744210 6625109

First 20 Pell-Lucas: 2 2 6 14 34 82 198 478 1154 2786 6726 16238 39202 94642 228486 551614 1331714 3215042 7761798 18738638

First 20 rational approximations of sqrt(2) (1.4142135623730951):
1/1 ~= 1
3/2 ~= 1.5
7/5 ~= 1.4
17/12 ~= 1.416666666666667
41/29 ~= 1.413793103448276
99/70 ~= 1.414285714285714
239/169 ~= 1.414201183431953
577/408 ~= 1.41421568627451
1393/985 ~= 1.414213197969543
3363/2378 ~= 1.41421362489487
8119/5741 ~= 1.414213551646055
19601/13860 ~= 1.414213564213564
47321/33461 ~= 1.41421356205732
114243/80782 ~= 1.414213562427273
275807/195025 ~= 1.414213562363799
665857/470832 ~= 1.41421356237469
1607521/1136689 ~= 1.414213562372821
3880899/2744210 ~= 1.414213562373142
9369319/6625109 ~= 1.414213562373087
22619537/15994428 ~= 1.414213562373097

First 20 Pell primes:
2
5
29
5741
33461
44560482149
1746860020068409
68480406462161287469
13558774610046711780701
4125636888562548868221559797461449
4760981394323203445293052612223893281
161733217200188571081311986634082331709
29647935552727996719...32712780937764687561 (64 digits)
67741382025708508432...87756429585398537901 (69 digits)
45562852543334487715...91808671945330235841 (73 digits)
54971607658948646301...49971207793263738989 (200 digits)
14030291214037674827...9886357032327271649 (356 digits)
24348043146521993819...3590369653438042689 (466 digits)
34643489561492944482...62121115635939797709 (498 digits)
32074710952523740376...14678081652481281009 (521 digits)

Indices of first 20 Pell primes: 2 3 5 11 13 29 41 53 59 89 97 101 167 181 191 523 929 1217 1301 1361

First 20 Newman-Shank-Williams numbers: 1 7 41 239 1393 8119 47321 ... 72722761475561 423859315570607

First 20 near isosceles right triangles:
[3, 4, 5]
[20, 21, 29]
[119, 120, 169]
[696, 697, 985]
[4059, 4060, 5741]
[23660, 23661, 33461]
[137903, 137904, 195025]
[803760, 803761, 1136689]
[4684659, 4684660, 6625109]
[27304196, 27304197, 38613965]
[159140519, 159140520, 225058681]
[927538920, 927538921, 1311738121]
[5406093003, 5406093004, 7645370045]
[31509019100, 31509019101, 44560482149]
[183648021599, 183648021600, 259717522849]
[1070379110496, 1070379110497, 1513744654945]
[6238626641379, 6238626641380, 8822750406821]
[36361380737780, 36361380737781, 51422757785981]
[211929657785303, 211929657785304, 299713796309065]
[1235216565974040, 1235216565974041, 1746860020068409]

Raku[edit]

my $pell = cache lazy 0, 1, * + * × 2*;
my $pell-lucas = lazy 2, 2, * + * × 2*;
 
my $upto = 20;
 
say "First $upto Pell numbers:\n" ~ $pell[^$upto];
 
say "\nFirst $upto Pell-Lucas numbers:\n" ~ $pell-lucas[^$upto];
 
say "\nFirst $upto rational approximations of √2 ({sqrt(2)}):\n" ~
(1..$upto).map({ sprintf "%d/%d - %1.16f", $pell[$_-1] + $pell[$_], $pell[$_], ($pell[$_-1]+$pell[$_])/$pell[$_] }).join: "\n";
 
say "\nFirst $upto Pell primes:\n" ~ $pell.grep(&is-prime)[^$upto].join: "\n";
 
say "\nIndices of first $upto Pell primes:\n" ~ (^).grep({$pell[$_].is-prime})[^$upto];
 
say "\nFirst $upto Newman-Shank-Williams numbers:\n" ~ (^$upto).map({ $pell[2 × $_, 2 × $_+1].sum });
 
say "\nFirst $upto near isosceles right triangles:";
map -> \p { printf "(%d, %d, %d)\n", |($_, $_+1 given $pell[^(2 × p + 1)].sum), $pell[2 × p + 1] }, 1..$upto;
Output:
First 20 Pell numbers:
0 1 2 5 12 29 70 169 408 985 2378 5741 13860 33461 80782 195025 470832 1136689 2744210 6625109

First 20 Pell-Lucas numbers:
2 2 6 14 34 82 198 478 1154 2786 6726 16238 39202 94642 228486 551614 1331714 3215042 7761798 18738638

First 20 rational approximations of √2 (1.4142135623730951):
1/1 - 1.0000000000000000
3/2 - 1.5000000000000000
7/5 - 1.4000000000000000
17/12 - 1.4166666666666667
41/29 - 1.4137931034482758
99/70 - 1.4142857142857144
239/169 - 1.4142011834319526
577/408 - 1.4142156862745099
1393/985 - 1.4142131979695431
3363/2378 - 1.4142136248948696
8119/5741 - 1.4142135516460548
19601/13860 - 1.4142135642135643
47321/33461 - 1.4142135620573204
114243/80782 - 1.4142135624272734
275807/195025 - 1.4142135623637995
665857/470832 - 1.4142135623746899
1607521/1136689 - 1.4142135623728214
3880899/2744210 - 1.4142135623731420
9369319/6625109 - 1.4142135623730870
22619537/15994428 - 1.4142135623730965

First 20 Pell primes:
2
5
29
5741
33461
44560482149
1746860020068409
68480406462161287469
13558774610046711780701
4125636888562548868221559797461449
4760981394323203445293052612223893281
161733217200188571081311986634082331709
2964793555272799671946653940160950323792169332712780937764687561
677413820257085084326543915514677342490435733542987756429585398537901
4556285254333448771505063529048046595645004014152457191808671945330235841
54971607658948646301386783144964782698772613513307493180078896702918825851648683235325858118170150873214978343601463118106546653220435805362395962991295556488036606954237309847762149971207793263738989
14030291214037674827921599320400561033992948898216351802670122530401263880575255235196727095109669287799074570417579539629351231775861429098849146880746524269269235328805333087546933690012894630670427794266440579064751300508834822795162874147983974059159392260220762973563561382652223360667198516093199367134903695783143116067743023134509886357032327271649
2434804314652199381956027075145741187716221548707931096877274520825143228915116227412484991366386864484767844200542482630246332092069382947111767723898168035847078557798454111405556629400142434835890123610082763986456199467423944182141028870863302603437534363208996458153115358483747994095302552907353919742211197822912892578751357668345638404394626711701120567186348490247426710813709165801137112237291901437566040249805155494297005186344325519103590369653438042689
346434895614929444828445967916634653215454504812454865104089892164276080684080254746939261017687341632569935171059945916359539268094914543114024020158787741692287531903178502306292484033576487391159597130834863729261484555671037916432206867189514675750227327687799973497042239286045783392065227614939379139866240959756584073664244580698830046194724340448293320938108876004367449471918175071251610962540447986139876845105399212429593945098472125140242905536711601925585608153109062121115635939797709
32074710952523740376423283403256578238321646122759160107427497117576305397686814013623874765833543023397971470911301264845142006214276865917420065183527313421909784286074786922242104480428021290764613639424408361555091057197776876849282654018358993099016644054242247557103410808928387071991436781136646322261169941417916607548507224950058710729258466238995253184617782314756913932650536663800753256087990078866003788647079369825102832504351225446531057648755795494571534144773842019836572551455718577614678081652481281009

Indices of first 20 Pell primes:
2 3 5 11 13 29 41 53 59 89 97 101 167 181 191 523 929 1217 1301 1361

First 20 Newman-Shank-Williams numbers:
1 7 41 239 1393 8119 47321 275807 1607521 9369319 54608393 318281039 1855077841 10812186007 63018038201 367296043199 2140758220993 12477253282759 72722761475561 423859315570607

First 20 near isosceles right triangles:
(3, 4, 5)
(20, 21, 29)
(119, 120, 169)
(696, 697, 985)
(4059, 4060, 5741)
(23660, 23661, 33461)
(137903, 137904, 195025)
(803760, 803761, 1136689)
(4684659, 4684660, 6625109)
(27304196, 27304197, 38613965)
(159140519, 159140520, 225058681)
(927538920, 927538921, 1311738121)
(5406093003, 5406093004, 7645370045)
(31509019100, 31509019101, 44560482149)
(183648021599, 183648021600, 259717522849)
(1070379110496, 1070379110497, 1513744654945)
(6238626641379, 6238626641380, 8822750406821)
(36361380737780, 36361380737781, 51422757785981)
(211929657785303, 211929657785304, 299713796309065)
(1235216565974040, 1235216565974041, 1746860020068409)

Scala[edit]

val pellNumbers: LazyList[BigInt] =
BigInt("0") #:: BigInt("1") #::
(pellNumbers zip pellNumbers.tail).
map{ case (a,b) => 2*b + a }
 
val pellLucasNumbers: LazyList[BigInt] =
BigInt("2") #:: BigInt("2") #::
(pellLucasNumbers zip pellLucasNumbers.tail).
map{ case (a,b) => 2*b + a }
 
val pellPrimes: LazyList[BigInt] =
pellNumbers.tail.tail.
filter{ case p => p.isProbablePrime(16) }
 
val pellIndexOfPrimes: LazyList[BigInt] =
pellNumbers.tail.tail.
filter{ case p => p.isProbablePrime(16) }.
map{ case p => pellNumbers.indexOf(p) }
 
val pellNSWnumbers: LazyList[BigInt] =
(pellNumbers.zipWithIndex.collect{ case (p,i) if i%2 == 0 => p}
zip
pellNumbers.zipWithIndex.collect{ case (p,i) if i%2 != 0 => p}).
map{ case (a,b) => a + b }
 
val pellSqrt2Numerator: LazyList[BigInt] =
BigInt(1) #:: BigInt(3) #::
(pellSqrt2Numerator zip pellSqrt2Numerator.tail).
map{ case (a,b) => 2*b + a }
 
val pellSqrt2: LazyList[BigDecimal] =
(pellSqrt2Numerator zip pellNumbers.tail).
map{ case (n,d) => BigDecimal(n)/BigDecimal(d) }
 
val pellSqrt2asString: LazyList[String] =
(pellSqrt2Numerator zip pellNumbers.tail).
map{ case (n,d) => s"$n/$d" }
 
val pellHypotenuse: LazyList[BigInt] =
pellNumbers.tail.tail.zipWithIndex.collect{ case (p,i) if i%2 != 0 => p }
 
val pellShortLeg: LazyList[BigInt] =
LazyList.from(3,2).map{ case s => pellNumbers.take(s).sum }
 
val pellTriple: LazyList[(BigInt,BigInt,BigInt)] =
(pellHypotenuse zip pellShortLeg).
map{ case (h,s) => (s,s+1,h)}
 
// Output
{ println("7 Tasks")
println("-------")
println(pellNumbers.take(10).mkString("1. Pell Numbers: ", ",", "\n"))
println(pellLucasNumbers.take(10).mkString("2. Pell-Lucas Numbers: ", ",", "\n"))
println((pellSqrt2asString zip pellSqrt2).take(10).
map { case (f, d) => s"$f = $d" }.
mkString("3. Square-root of 2 Approximations: \n\n",
"\n", "\n"))
println(pellPrimes.take(10).mkString("4. Pell Primes: \n\n", "\n", "\n"))
println(pellIndexOfPrimes.take(10).mkString("5. Pell Index of Primes: ", ",", "\n"))
println(pellNSWnumbers.take(10).mkString("6. Newman-Shank-Williams Numbers: \n\n", "\n", "\n"))
println(pellTriple.take(10).mkString("7. Near Right-triangle Triples: \n\n", "\n", "\n"))
}
 
Output:
7 Tasks
-------
1. Pell Numbers: 0,1,2,5,12,29,70,169,408,985

2. Pell-Lucas Numbers: 2,2,6,14,34,82,198,478,1154,2786

3. Square-root of 2 Approximations:

1/1 = 1
3/2 = 1.5
7/5 = 1.4
17/12 = 1.416666666666666666666666666666667
41/29 = 1.413793103448275862068965517241379
99/70 = 1.414285714285714285714285714285714
239/169 = 1.414201183431952662721893491124260
577/408 = 1.414215686274509803921568627450980
1393/985 = 1.414213197969543147208121827411168
3363/2378 = 1.414213624894869638351555929352397

4. Pell Primes:

2
5
29
5741
33461
44560482149
1746860020068409
68480406462161287469
13558774610046711780701
4125636888562548868221559797461449

5. Pell Index of Primes: 2,3,5,11,13,29,41,53,59,89

6. Newman-Shank-Williams Numbers:

1
7
41
239
1393
8119
47321
275807
1607521
9369319

7. Near Right-triangle Triples:

(3,4,5)
(20,21,29)
(119,120,169)
(696,697,985)
(4059,4060,5741)
(23660,23661,33461)
(137903,137904,195025)
(803760,803761,1136689)
(4684659,4684660,6625109)
(27304196,27304197,38613965)

Wren[edit]

Library: Wren-big
Library: Wren-math
Library: Wren-fmt
import "./big" for BigInt, BigRat
import "./math" for Int
import "./fmt" for Fmt
 
var p = List.filled(40, 0)
p[0] = 0
p[1] = 1
for (i in 2..39) p[i] = 2 * p[i-1] + p[i-2]
System.print("The first 20 Pell numbers are:")
System.print(p[0..19].join(" "))
 
var q = List.filled(40, 0)
q[0] = 2
q[1] = 2
for (i in 2..39) q[i] = 2 * q[i-1] + q[i-2]
System.print("\nThe first 20 Pell-Lucas numbers are:")
System.print(q[0..19].join(" "))
 
System.print("\nThe first 20 rational approximations of √2 (1.4142135623730951) are:")
for (i in 1..20) {
var r = BigRat.new(q[i]/2, p[i])
Fmt.print("$-17s ≈ $-18s", r, r.toDecimal(16, true, true))
}
 
System.print("\nThe first 15 Pell primes are:")
var p0 = BigInt.zero
var p1 = BigInt.one
var indices = List.filled(15, 0)
var count = 0
var index = 2
var p2
while (count < 15) {
p2 = p1 * BigInt.two + p0
if (Int.isPrime(index) && p2.isProbablePrime(10)) {
System.print(p2)
indices[count] = index
count = count + 1
}
index = index + 1
p0 = p1
p1 = p2
}
 
System.print("\nIndices of the first 15 Pell primes are:")
System.print(indices.join(" "))
 
System.print("\nFirst 20 Newman-Shank_Williams numbers:")
var nsw = List.filled(20, 0)
for (n in 0..19) nsw[n] = p[2*n] + p[2*n+1]
Fmt.print("$d", nsw)
 
System.print("\nFirst 20 near isosceles right triangles:")
p0 = 0
p1 = 1
var sum = 1
var i = 2
while (i < 43) {
p2 = p1 * 2 + p0
if (i % 2 == 1) {
Fmt.print("($d, $d, $d)", sum, sum + 1, p2)
}
sum = sum + p2
p0 = p1
p1 = p2
i = i + 1
}
Output:
The first 20 Pell numbers are:
0 1 2 5 12 29 70 169 408 985 2378 5741 13860 33461 80782 195025 470832 1136689 2744210 6625109

The first 20 Pell-Lucas numbers are:
2 2 6 14 34 82 198 478 1154 2786 6726 16238 39202 94642 228486 551614 1331714 3215042 7761798 18738638

The first 20 rational approximations of √2 (1.4142135623730951) are:
1/1               ≈ 1.0000000000000000
3/2               ≈ 1.5000000000000000
7/5               ≈ 1.4000000000000000
17/12             ≈ 1.4166666666666667
41/29             ≈ 1.4137931034482759
99/70             ≈ 1.4142857142857143
239/169           ≈ 1.4142011834319527
577/408           ≈ 1.4142156862745098
1393/985          ≈ 1.4142131979695431
3363/2378         ≈ 1.4142136248948696
8119/5741         ≈ 1.4142135516460547
19601/13860       ≈ 1.4142135642135642
47321/33461       ≈ 1.4142135620573205
114243/80782      ≈ 1.4142135624272734
275807/195025     ≈ 1.4142135623637995
665857/470832     ≈ 1.4142135623746899
1607521/1136689   ≈ 1.4142135623728214
3880899/2744210   ≈ 1.4142135623731420
9369319/6625109   ≈ 1.4142135623730870
22619537/15994428 ≈ 1.4142135623730964

The first 15 Pell primes are:
2
5
29
5741
33461
44560482149
1746860020068409
68480406462161287469
13558774610046711780701
4125636888562548868221559797461449
4760981394323203445293052612223893281
161733217200188571081311986634082331709
2964793555272799671946653940160950323792169332712780937764687561
677413820257085084326543915514677342490435733542987756429585398537901
4556285254333448771505063529048046595645004014152457191808671945330235841

Indices of the first 15 Pell primes are:
2 3 5 11 13 29 41 53 59 89 97 101 167 181 191

First 20 Newman-Shank_Williams numbers:
1 7 41 239 1393 8119 47321 275807 1607521 9369319 54608393 318281039 1855077841 10812186007 63018038201 367296043199 2140758220993 12477253282759 72722761475561 423859315570607

First 20 near isosceles right triangles:
(3, 4, 5)
(20, 21, 29)
(119, 120, 169)
(696, 697, 985)
(4059, 4060, 5741)
(23660, 23661, 33461)
(137903, 137904, 195025)
(803760, 803761, 1136689)
(4684659, 4684660, 6625109)
(27304196, 27304197, 38613965)
(159140519, 159140520, 225058681)
(927538920, 927538921, 1311738121)
(5406093003, 5406093004, 7645370045)
(31509019100, 31509019101, 44560482149)
(183648021599, 183648021600, 259717522849)
(1070379110496, 1070379110497, 1513744654945)
(6238626641379, 6238626641380, 8822750406821)
(36361380737780, 36361380737781, 51422757785981)
(211929657785303, 211929657785304, 299713796309065)
(1235216565974040, 1235216565974041, 1746860020068409)