Pell numbers: Difference between revisions

Content added Content deleted

Inline

Revision as of 16:20, 5 March 2022

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;
P_n = 2 × P_n-1 + P_n-2;

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


P_n = ((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;
Q_n = 2 × Q_n-1 + Q_n-2;

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


Q_n = (1 + √2)ⁿ + (1 - √2)ⁿ;

or


Q_n = P_2n / P_n;

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 P_n and the numerator is Q_n / 2 or P_n-1 + P_n.

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 number.

If you take the sum S of the first 4n + 1 Pell numbers, the sum of the terms P_2n and P_{2n + 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 P_2n + P_{2n + 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 P_{2n + 1}. The shorter leg length is the sum of the terms up to P_{2n + 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

OEIS:A000129 - Pell numbers OEIS:A002203 - Companion Pell numbers (Pell-Lucas numbers) OEIS:A001333 - Numerators of continued fraction convergents to sqrt(2) (Companion Pell numbers / 2) OEIS:A086383 - Prime terms in the sequence of Pell numbers OEIS:A096650 - Indices of prime Pell numbers OEIS:A002315 - NSW numbers (Newman-Shank-Williams numbers) Wikipedia: Pythagorean_triple

Raku

<lang perl6>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 $upto near isoceles right tringles:"; map -> \p { printf "(%d, %d, %d)\n", |($_, $_+1 given $pell[^(2 × p + 1)].sum), $pell[2 × p + 1] }, 1..$upto;</lang>

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 isoceles right tringles:
(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)

@@ Line 1: / Line 1: @@
 {{draft task}}
-'''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 manuy other interesting uses and relationships.
+'''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'''.