Pell numbers: Difference between revisions

=={{header|ALGOL 68}}==

{{Trans|FreeBASIC|using Millar Rabin for finding Pell primes and some minor output format differences and showing 10 Pell primes}}

Assumes LONG INT is large enough to hold the 90th Pell number (as in Algol 68G, versions 2 and 3).

{{libheader|ALGOL 68-primes}}

<syntaxhighlight lang="algol68">

BEGIN # find some Pell numbers - trans FreeBASIC ( which is trans Phix ) #

PR read "primes.incl.a68" PR

PROC is prime = ( LONG INT v )BOOL:

IF v < 2 THEN FALSE

ELIF v MOD 2 = 0 THEN v = 2

ELIF v MOD 3 = 0 THEN v = 3

ELSE

LONG INT d := 5;

BOOL result := TRUE;

WHILE result AND d * d <= v DO

IF v MOD d = 0

THEN result := FALSE

ELSE

d +:= 2;

IF v MOD d = 0

THEN result := FALSE

ELSE d +:= 4

FI

FI

OD;

result

FI # is prime # ;

INT n;

[ 0 : 90 ]LONG INT p, pl;

p[ 0 ] := 0; p[ 1 ] := 1;

pl[ 0 ] := 2; pl[ 1 ] := 2;

FOR n FROM 2 TO UPB p DO

p[ n ] := 2 * p[ n - 1 ] + p[ n - 2 ];

pl[ n ] := 2 * pl[ n - 1 ] + pl[ n - 2 ]

OD;

print( ( "First 20 Pell numbers:", newline ) );

FOR n FROM 0 TO 19 DO print( ( " ", whole( p[ n ], 0 ) ) ) OD;

print( ( newline, newline, "First 20 Pell-Lucas numbers:", newline ) );

FOR n FROM 0 TO 19 DO print( ( " ", whole( pl[ n ], 0 ) ) ) OD;

print( ( newline, newline, "First 20 rational approximations of sqrt(2) (" ) );

print( ( fixed( sqrt( 2 ), -15, 13 ), "):", newline ) );

FOR n TO 20 DO

LONG INT j = pl[ n ] OVER 2, d = p[ n ];

print( ( " ", whole( j, 0 ), "/", whole( d, 0 ), " ~= ", fixed( j / d, -15, 13 ), newline ) )

OD;

print( ( newline, "First 10 Pell primes:", newline, "index Pell prime", newline ) );

INT pdx := 2;

INT c := 0;

WHILE

IF is probably prime( p[ pdx ] ) THEN

print( ( whole( pdx, -5 ), " ", whole( p[ pdx ], 0 ), newline ) );

c +:= 1

FI;

pdx +:= 1;

c < 10

DO SKIP OD;

[ 0 : 20 ]LONG INT nsw;

FOR n FROM 0 TO 19 DO nsw[ n ] := p[ 2 * n ] + p[ 2 * n + 1 ] OD;

print( ( newline, newline, "First 20 Newman-Shank-Williams numbers:", newline ) );

FOR n FROM 0 TO 19 DO print( ( " ", whole( nsw[ n ], 0 ) ) ); IF n = 13 THEN print( ( newline ) ) FI OD;

print( ( newline, newline, "First 20 near isosceles right triangles:", newline ) );

LONG INT i0 := 0, i1 := 1, t := 1, found := 0;

FOR i FROM 2 WHILE found < 20 DO

LONG INT i2 = i1*2 + i0;

IF ODD i THEN

print( ( " [", whole( t, 0 ), ", ", whole( t + 1, 0 ), ", ", whole( i2, 0 ), "]", newline ) );

found +:= 1

FI;

t +:= i2; i0 := i1; i1 := i2

OD

END

</syntaxhighlight>

{{out}}

<pre style="height:40ex;overflow:scroll;">

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 sqrt(2) (1.4142135623731):

1/1 ~= 1.0000000000000

3/2 ~= 1.5000000000000

7/5 ~= 1.4000000000000

17/12 ~= 1.4166666666667

41/29 ~= 1.4137931034483

99/70 ~= 1.4142857142857

239/169 ~= 1.4142011834320

577/408 ~= 1.4142156862745

1393/985 ~= 1.4142131979695

3363/2378 ~= 1.4142136248949

8119/5741 ~= 1.4142135516461

19601/13860 ~= 1.4142135642136

47321/33461 ~= 1.4142135620573

114243/80782 ~= 1.4142135624273

275807/195025 ~= 1.4142135623638

665857/470832 ~= 1.4142135623747

1607521/1136689 ~= 1.4142135623728

3880899/2744210 ~= 1.4142135623731

9369319/6625109 ~= 1.4142135623731

22619537/15994428 ~= 1.4142135623731

First 10 Pell primes:

index Pell prime

2 2

3 5

5 29

11 5741

13 33461

29 44560482149

41 1746860020068409

53 68480406462161287469

59 13558774610046711780701

89 4125636888562548868221559797461449

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]

</pre>