# Largest palindrome product

Largest palindrome product is a draft programming task. It is not yet considered ready to be promoted as a complete task, for reasons that should be found in its talk page.

Task description is taken from Project Euler (https://projecteuler.net/problem=4)
A palindromic number reads the same both ways. The largest palindrome made from the product of two 2-digit numbers is 9009 = 91 × 99.
Find the largest palindrome made from the product of two 3-digit numbers.

Stretch Goal

Find the largest palindrome made from the product of two n-digit numbers, where n ranges from 4 to 7.

Extended Stretch Goal

Find the largest palindrome made from the product of two n-digit numbers, where n ranges beyond 7,

## 11l

Translation of: Wren
```F reverse(=n)
V r = Int64(0)
L n > 0
r = n % 10 + r * 10
n I/= 10
R r

V po = Int64(10)
L(n) 2..7
V low = po * 9
po *= 10
V high = po - 1
V nextN = 0B
L(i) (high .. low).step(-1)
V j = reverse(i)
V p = i * po + j
V k = high
L k > low
I k % 10 != 5
V l = p I/ k
I l > high
L.break
I p % k == 0
print(‘Largest palindromic product of two ’n‘-digit integers: ’k‘ x ’l‘ = ’p)
nextN = 1B
L.break
k -= 2
I nextN
L.break```
Output:
```Largest palindromic product of two 2-digit integers: 99 x 91 = 9009
Largest palindromic product of two 3-digit integers: 993 x 913 = 906609
Largest palindromic product of two 4-digit integers: 9999 x 9901 = 99000099
Largest palindromic product of two 5-digit integers: 99979 x 99681 = 9966006699
Largest palindromic product of two 6-digit integers: 999999 x 999001 = 999000000999
Largest palindromic product of two 7-digit integers: 9998017 x 9997647 = 99956644665999
```

## ALGOL 68

Translation of: Wren
```BEGIN # find the highest palindromic multiple of various sizes of numbers #
# returns n with the digits reversed #
PROC reverse = ( LONG INT v )LONG INT:
BEGIN
LONG INT r := 0 ;
LONG INT n := v;
WHILE n > 0 DO
r *:= 10; r +:= n MOD 10;
n OVERAB 10
OD;
r
END # reverse # ;

LONG INT pow := 10;
FOR n FROM 2 TO 7 DO
LONG INT low := pow * 9;
pow *:= 10;
LONG INT high := pow - 1;
print( ( "Largest palindromic product of two ", whole( n, 0 ), "-digit integers: " ) );
BOOL next n := FALSE;
LONG INT i := high + 1;
WHILE i -:= 1;
i >= low AND NOT next n
DO
LONG INT j = reverse( i );
LONG INT p = ( i * pow ) + j;
# k can't be even nor end in 5 to produce a product ending in 9 #
LONG INT k := high + 2;
WHILE k -:= 2;
IF   k < low
THEN FALSE
ELIF k MOD 10 = 5
THEN TRUE
ELIF LONG INT l = p OVER k;
l > high
THEN FALSE
ELIF p MOD k = 0 THEN
print( ( whole( k, 0 ), " x ", whole( l, 0 ), " = ", whole( p, 0 ), newline ) );
next n := TRUE;
FALSE
ELSE TRUE
FI
DO SKIP OD
OD
OD
END```
Output:
```Largest palindromic product of two 2-digit integers: 99 x 91 = 9009
Largest palindromic product of two 3-digit integers: 993 x 913 = 906609
Largest palindromic product of two 4-digit integers: 9999 x 9901 = 99000099
Largest palindromic product of two 5-digit integers: 99979 x 99681 = 9966006699
Largest palindromic product of two 6-digit integers: 999999 x 999001 = 999000000999
Largest palindromic product of two 7-digit integers: 9998017 x 9997647 = 99956644665999
```
Translation of: Ring

Also showing the maximum for 2 and 4 .. 7 digit numbers. Tests for a better product before testing for palindromicity.

```BEGIN # find the highest palindromic multiple of various sizes of numbers #
PROC is pal = ( LONG INT n )BOOL:
BEGIN
STRING x       = whole( n, 0 );
INT    l      := UPB x + 1;
BOOL   result := TRUE;
FOR i FROM LWB x WHILE i < l AND result DO
l     -:= 1;
result := x[ i ] = x[ l ]
OD;
result
END # is pal # ;

# maximum 2 digit number #
LONG INT max := 99;
# both factors must be >= 10for a 4 digit product #
LONG INT limit start := 10;
FOR w FROM 2 TO 7 DO
LONG INT best prod   := 0;
# one factor must be divisible by 11 #
LONG INT limit end = 11 * ( max OVER 11 );
LONG INT second   := limit start;
LONG INT first    := 1;
# loop from hi to low to find the best result in the fewest steps #
LONG INT n        := limit end + 11;
WHILE n -:= 11;
n >= limit start
DO
# with n falling, the lower limit of m can rise with #
# the best-found-so-far second number. Doing this #
# lowers the iteration count by a lot. #
LONG INT m := max + 2;
WHILE m -:= 2;
IF   m < second
THEN FALSE
ELIF LONG INT prod = n * m;
best prod > prod
THEN FALSE
ELIF NOT is pal( prod )
THEN TRUE
ELSE # maintain the best-found-so-far result #
first     := n;
second    := m;
best prod := prod;
TRUE
FI
DO SKIP OD
OD;
print( ( "Largest palindromic product of two ", whole( w, 0 )
, "-digit numbers: ", whole( first, 0 ), " * ", whole( second, 0 )
, " = ", whole( best prod, 0 )
, newline
)
);
max *:= 10;
max +:= 9;
limit start *:= 10
OD
END```
Output:
```Largest palindromic product of two 2-digit numbers: 99 * 91 = 9009
Largest palindromic product of two 3-digit numbers: 913 * 993 = 906609
Largest palindromic product of two 4-digit numbers: 9999 * 9901 = 99000099
Largest palindromic product of two 5-digit numbers: 99979 * 99681 = 9966006699
Largest palindromic product of two 6-digit numbers: 999999 * 999001 = 999000000999
Largest palindromic product of two 7-digit numbers: 9997647 * 9998017 = 99956644665999
```

## Arturo

```palindrome?: function [n]->
(to :string n) = reverse to :string n

getAllMuls: function [n][
result: []
limFrom: 10^ n-1
limTo: dec 10^n
loop limFrom..limTo 'a [
loop limFrom..limTo 'b [
m: a*b
if palindrome? m ->
'result ++ @[@[a, b, a*b]]
]
]
return result
]

largestPal: maximum getAllMuls 3 'x -> last x
print ["Largest palindromic product of two 3-digit integers:" largestPal\0 "x" largestPal\1 "=" largestPal\2]
```
Output:
`Largest palindromic product of two 3-digit integers: 913 x 993 = 906609`

## AWK

```# syntax: GAWK -f LARGEST_PALINDROME_PRODUCT.AWK
BEGIN {
main(9)
main(99)
main(999)
main(9999)
exit(0)
}
function main(n,  i,j,max_i,max_j,max_product,product) {
for (i=1; i<=n; i++) {
for (j=1; j<=n; j++) {
product = i * j
if (product > max_product) {
if (product ~ /^9/ && product ~ /9\$/) {
if (product == reverse(product)) {
max_product = product
max_i = i
max_j = j
}
}
}
}
}
printf("%1d: %4s * %-4s = %d\n",length(n),max_i,max_j,max_product)
}
function reverse(str,  i,rts) {
for (i=length(str); i>=1; i--) {
rts = rts substr(str,i,1)
}
return(rts)
}
```
Output:
```1:    1 * 9    = 9
2:   91 * 99   = 9009
3:  913 * 993  = 906609
4: 9901 * 9999 = 99000099
```

## Ksh

```#!/bin/ksh

# Largest palindrome product of two 3-digit numbers

#	# Variables:
#
typeset -si MINFACT=913		# From 'Paper & Pencil' solution
typeset -si MAXFACT=999

#	# Functions:
#

#	# Function _ispalindrome(n) - return 1 for palindromic number
#
function _ispalindrome {
typeset _n ; integer _n="\$1"

(( _n != \$(_flipit \${_n}) )) && return 0
return 1
}

#	# Function _flipit(string) - return flipped string
#
function _flipit {
typeset _buf ; _buf="\$1"
typeset _tmp ; unset _tmp
typeset _i ; typeset -si _i

for (( _i=\$(( \${#_buf}-1 )); _i>=0; _i-- )); do
_tmp="\${_tmp}\${_buf:\${_i}:1}"
done
echo "\${_tmp}"
}

######
# main #
######

integer prod MAXPPROD=0
for (( i=MINFACT; i<=MAXFACT; i++)); do
for (( j=MINFACT; j<=MAXFACT; j++)); do
(( prod = i * j ))
_ispalindrome \${prod}
(( \$? )) && (( prod > MAXPPROD )) && MAXPPROD=\${prod}
done
done

print "Largest palindrome product of two 3-digit factors = \${MAXPPROD}"
```
Output:
```
Largest palindrome product of two 3-digit factors = 906609```

## C#

Translation of: Paper & Pencil
```using System;
class Program {

static bool isPal(int n) {
int rev = 0, lr = -1, rem;
while (n > rev) {
n = Math.DivRem(n, 10, out rem);
if (lr < 0 && rem == 0) return false;
lr = rev; rev = 10 * rev + rem;
if (n == rev || n == lr) return true;
} return false; }

static void Main(string[] args) {
var sw = System.Diagnostics.Stopwatch.StartNew();
int x = 900009, y = (int)Math.Sqrt(x), y10, max = 999, max9 = max - 9, z, p, bp = x, ld, c;
var a = new int[]{ 0,9,0,3,0,0,0,7,0,1 }; string bs = "";
y /= 11;
if ((y & 1) == 0) y--;
if (y % 5 == 0) y -= 2;
y *= 11;
while (y <= max) {
c = 0;
y10 = y * 10;
z = max9 + a[ld = y % 10];
p = y * z;
while (p >= bp) {
if (isPal(p)) {
if (p > bp) bp = p;
bs = string.Format("{0} x {1} = {2}", y, z - c, bp);
}
p -= y10; c += 10;
}
y += ld == 3 ? 44 : 22;
}
sw.Stop();
Console.Write("{0} {1} μs", bs, sw.Elapsed.TotalMilliseconds * 1000.0);
}
}
```
Output @ Tio.run:
`913 x 993 = 906609 245.2 μs`

### Stretch

```using System;

class Program {

static bool isPal(long n) {
long rev = 0, lr = -1, rem;
while (n > rev) {
n = Math.DivRem(n, 10, out rem);
if (lr < 0 && rem == 0) return false;
lr = rev; rev = 10 * rev + rem;
if (n == rev || n == lr) return true;
} return false; }

static void doOne(int n) {
int ld, c; string bs = "";
string sx = "9" + new string('0', (n - 1) << 1) + "9", sm = new string('9', n);
long x = long.Parse(sx), y = (long)Math.Sqrt(x), oy, max = long.Parse(sm), max9 = max - 9, z, yy, p, bp = x;
var a = new long[] { 0, 9, 0, 3, 0, 0, 0, 7, 0, 1 };
y /= 11;
if ((y & 1) == 0) y--;
if (y % 5 == 0) y -= 2;
y *= 11; oy = y;
while (y <= max) y += 22; y -= 22;
while (y >= oy) {
c = 0;
yy = y * 10;
z = max9 + a[ld = (int)(y % 10)];
//Console.WriteLine("y,z: {0},{1}", y, z);
p = y * z;
while (p >= bp) {
if (isPal(p)) {
if (p > bp) bp = p;
bs = string.Format(" {0,2} {1,10} x {2,-10} = {3}{4}", n, y, z - c, new string(' ', 10 - n), bp); }
p -= yy; c += 10; }
y -= ld == 7 ? 44 : 22; }
Console.WriteLine(bs); }

static void Main(string[] args) {
Console.WriteLine("digs    factor   factor            palindrome");
var sw = System.Diagnostics.Stopwatch.StartNew();
for (int i = 2, h = 1; i <= 10; h = ++i >> 1) {
if ((i & 1) == 0) {
string b = new string('9', i),
a = new string('9', h) + new string('0', (h) - 1) + "1",
c = new string('9', h) + new string('0', i) + new string('9', h);
Console.WriteLine(" {0,2} {1,10} x {2,-10} = {3}{4}", i, a, b, new string(' ', 10 - i), c); }
else doOne(i);
}
sw.Stop();
Console.Write("{0} sec", sw.Elapsed.TotalSeconds);
}
}
```
Output @ Tio.run:

Showing results for 2 through 10 digit factors.

```digs    factor   factor            palindrome
2         91 x 99         =         9009
3        913 x 993        =        906609
4       9901 x 9999       =       99000099
5      99979 x 99681      =      9966006699
6     999001 x 999999     =     999000000999
7    9997647 x 9998017    =    99956644665999
8   99990001 x 99999999   =   9999000000009999
9  999920317 x 999980347  =  999900665566009999
10 9999900001 x 9999999999 = 99999000000000099999
2.1622142 sec```
Wow! how did that go so fast? The results for the even-number-of-digit factors were manufactured by string manipulation instead of calculation (since the pattern was obvious). This algorithm can easily be adapted to BigIntegers for higher n-digit factors, but the execution time is unspectacular.

## Delphi

Works with: Delphi version 6.0

```type TProgress = procedure(Percent: integer);

function ReverseNum(N: int64): int64;
{Reverse the digit order of a number}
begin
Result:=0;
while N>0 do
begin
Result:=(Result*10)+(N mod 10);
N:=N div 10;
end;
end;

function IsPalindrome(N: int64): boolean;
{If the number is the same in }
{reverse order it is a palindrome}
var N1: int64;
begin
N1:=ReverseNum(N);
Result:=N = N1;
end;

procedure ShowPalindrome(Memo: TMemo; D,N1,N2: int64);
begin
Memo.Lines.Add(Format('%5D  %5D  %5D  %10D',[D,N1,N2,N1 * N2]));
end;

procedure FindPalindromes(Digits: integer; var C1,C2: int64; Prog: TProgress);
{Find the largest palindrome derrived from two}
{ terms with the specified number of digits}
var I,J: cardinal;
var Prd,MinNum,MaxNum,Best: int64;
begin
Best:=0;
{Find the minimum and maximum values}
{ with the specified number of digits}
MinNum:=Trunc(Power(10,Digits-1));
MaxNum:=Trunc(Power(10,Digits))-1;

for I:=MinNum to MaxNum do
begin
{We can eliminate even factors and number ending in 5}
if ((I and 1)=0) or ((I mod 10)=5) then continue;
for J:=I+1 to MaxNum do
begin
{We can eliminate even factors and number ending in 5}
if ((J and 1)=0) or ((J mod 10)=5) then continue;
Prd:=I * J;
if not IsPalindrome(Prd) then continue;
if Assigned(Prog) then Prog(MulDiv(100,I-MinNum,MaxNum-MinNum));
{Save the largest palindromes}
if Prd>Best then
begin
Best:=Prd;
C1:=I; C2:=J;
end;
end;
end;
end;

procedure FindPalindromeMax(Memo: TMemo; Prog: TProgress);
var N1,N2: Int64;
var I: integer;
begin
for I:=2 to 4 do
begin
FindPalindromes(I,N1,N2,Prog);
ShowPalindrome(Memo,I,N1,N2);
end;
end;
```
Output:
```Digits    F1     F2  Palindrome
-------------------------------
2     91     99        9009
3    913    993      906609
4   9901   9999    99000099
```

## F#

```// Largest palindrome product. Nigel Galloway: November 3rd., 2021
let fN g=let rec fN g=[yield g%10; if g>=10 then yield! fN(g/10)] in let n=fN g in n=List.rev n
printfn "%d" ([for n in 100..999 do for g in n..999->n*g]|>List.filter fN|>List.max)
```
Output:
```906609
```

## FreeBASIC

### Version 1

```function make_pal( n as ulongint ) as ulongint
'turn a number into a palindrom with twice as many digits
dim as string ns, ret
ns = str(n) : ret = ns
for i as uinteger = len(ns) to 1 step -1
ret += mid(ns, i, 1)
next i
return val(ret)
end function

function has_dig( n as ulongint, d as uinteger ) as boolean
'does the number n have d decimal digits?
if 10^(d-1)<=n and n<10^d then return true else return false
end function

dim as integer np

for d as uinteger = 2 to 7
for n as ulongint = 10^d - 1 to 10^(d-1) step -1 'count down from 999...
'since the first good number we encounter
'must be the highest
np = make_pal( n )                           'produce a 2d-digit palindrome from it
for f as ulongint = 10^d - 1 to 10^(d-1) step -1   'look for highest d-digit factor
if np mod f = 0 then
if has_dig( np/f, d ) then           'if np/f also has d digits we are done :)
print f;" *";np/f;" =";np
goto nextd
end if
end if
next f
next n
nextd:                                           'yes, I used a goto. sue me.
next d```
Output:
```99 * 91 = 9009
993 * 913 = 906609
9999 * 9901 = 99000099
99979 * 99681 = 9966006699
999999 * 999001 = 999000000999
9998017 * 9997647 = 99956644665999```

### Version 2

This version is based on Version 1 with only a few changes and some extra code based on the fact that one divisor can be divided by 11, this speeds it even more up and a option for using goto, exit or continue. highest n is 9, (highest possible for unsigned 64bit integers).

```' version 07-10-2021
' compile with: fbc -s console

' Now you can choice, no speed changes for all 3
' 1: use goto
' 2: use exit
' 3: use continue
#Define Option_ 1  ' set option_ to 1, 2 or 3. for all other value's uses 1

Function make_pal( n As UInteger ) As ULongInt
'turn a number into a palindrom with twice as many digits
Dim As String ns = Str(n), ret = ns
For i As UInteger = Len(ns) To 1 Step -1
ret += Mid(ns, i, 1)
Next i
Return ValULng(ret)
End Function

Dim As ULongInt np, tmp
Dim As Double t1 =Timer
For d As UInteger = 2 To 9
For n As UInteger = 10^d -2 To 10^(d -1) Step -1
np = make_pal( n )
tmp = Sqr(np)
tmp = tmp - (10^d - 1 - tmp)
tmp = tmp - tmp Mod 11
If (tmp And 1) = 0 Then tmp = tmp + 11
For f As UInteger = tmp To 10^d -1  Step 22
If np Mod f = 0 Then
If np \ f > (10^d) Then Continue For
Print f; " * "; np \ f; " = "; np
#If (option_ = 2)
Exit For, For
#ElseIf (option_ = 3)
Continue For, For, For
#Else
GoTo nextd
#EndIf
End If
Next f
Next n
#If (option_ <> 2 Or option_ <> 3)
nextd:
#EndIf
Next d

Print Timer-t1
' empty keyboard buffer
While InKey <> "" : Wend
Print : Print "hit any key to end program"
Sleep
End```
Output:
```99 * 91 = 9009
993 * 913 = 906609
9999 * 9901 = 99000099
99979 * 99681 = 9966006699
999999 * 999001 = 999000000999
9998017 * 9997647 = 99956644665999
99999999 * 99990001 = 9999000000009999
999980347 * 999920317 = 999900665566009999```

## Go

Translation of: Wren

18 digit integers are within the range of Go's uint64 type though finding the result for 9-digit number products takes a while - around 15 seconds on my machine.

```package main

import "fmt"

func reverse(n uint64) uint64 {
r := uint64(0)
for n > 0 {
r = n%10 + r*10
n /= 10
}
return r
}

func main() {
pow := uint64(10)
nextN:
for n := 2; n < 10; n++ {
low := pow * 9
pow *= 10
high := pow - 1
fmt.Printf("Largest palindromic product of two %d-digit integers: ", n)
for i := high; i >= low; i-- {
j := reverse(i)
p := i*pow + j
// k can't be even nor end in 5 to produce a product ending in 9
for k := high; k > low; k -= 2 {
if k % 10 == 5 {
continue
}
l := p / k
if l > high {
break
}
if p%k == 0 {
fmt.Printf("%d x %d = %d\n", k, l, p)
continue nextN
}
}
}
}
}
```
Output:
```Largest palindromic product of two 2-digit integers: 99 x 91 = 9009
Largest palindromic product of two 3-digit integers: 993 x 913 = 906609
Largest palindromic product of two 4-digit integers: 9999 x 9901 = 99000099
Largest palindromic product of two 5-digit integers: 99979 x 99681 = 9966006699
Largest palindromic product of two 6-digit integers: 999999 x 999001 = 999000000999
Largest palindromic product of two 7-digit integers: 9998017 x 9997647 = 99956644665999
Largest palindromic product of two 8-digit integers: 99999999 x 99990001 = 9999000000009999
Largest palindromic product of two 9-digit integers: 999980347 x 999920317 = 999900665566009999
```

## jq

Works with: jq

Works with gojq, the Go implementation of jq

```def reverseNumber:
tostring|explode|reverse|implode|tonumber;

{ pow: 10}
| foreach range(2;8) as \$n (.;
(.pow * 9) as \$low
| .pow *= 10
| (.pow - 1) as \$high
| .emit = null
| .nextN = false
| label \$out
| foreach range(\$high; \$low - 1; -1) as \$i (.;
(\$i|reverseNumber) as \$j
| (\$i * .pow + \$j) as \$p
# k can't be even nor end in 5 to produce a product ending in 9
| .k = \$high
| .done = false
| until(.k <= \$low or .done;
if (.k % 10 != 5)
then (\$p / .k) as \$l
| if \$l > \$high
then .done = true
elif \$p % .k == 0
then .emit = "Largest palindromic product of two \(\$n)-digit integers: \(.k) x \(\$l) = \(\$p)"
| .nextN = true
| .done = true
else .
end
else .
end
| .k += -2 )
| if .nextN then ., break \$out else . end;
select(.emit) );
.emit ) ;

Output:
```Largest palindromic product of two 2-digit integers: 99 x 91 = 9009
Largest palindromic product of two 3-digit integers: 993 x 913 = 906609
Largest palindromic product of two 4-digit integers: 9999 x 9901 = 99000099
Largest palindromic product of two 5-digit integers: 99979 x 99681 = 9966006699
Largest palindromic product of two 6-digit integers: 999999 x 999001 = 999000000999
Largest palindromic product of two 7-digit integers: 9998017 x 9997647 = 99956644665999
```

## Julia

```using Primes

function twoprodpal(factorlength)
maxpal = Int128(10)^(2 * factorlength) - 1
dig = digits(maxpal)
halfnum = dig[1:length(dig)÷2]
while any(halfnum .!= 0)
prodnum = evalpoly(Int128(10), [reverse(halfnum); halfnum])
facs = twofac(factorlength, prodnum)
if !isempty(facs)
println("For factor length \$factorlength, \$(facs) * \$(facs) = \$prodnum")
break
end
halfnum = digits(evalpoly(Int128(10), halfnum) - 1)
end
end

function twofac(faclength, prodnum)
f = [one(prodnum)]
for (p, e) in factor(prodnum)
f = reduce(vcat, [f * p^j for j in 1:e], init=f)
end
possiblefacs = filter(x -> length(string(x)) == faclength, f)
for i in possiblefacs
j = prodnum ÷ i
j ∈ possiblefacs && return sort([i, j])
end
return typeof(prodnum)[]
end

twoprodpal(i)
end
```
Output:
```For factor length 2, 91 * 99 = 9009
For factor length 3, 913 * 993 = 906609
For factor length 4, 9901 * 9999 = 99000099
For factor length 5, 99681 * 99979 = 9966006699
For factor length 6, 999001 * 999999 = 999000000999
For factor length 7, 9997647 * 9998017 = 99956644665999
For factor length 8, 99990001 * 99999999 = 9999000000009999
For factor length 9, 999920317 * 999980347 = 999900665566009999
For factor length 10, 9999986701 * 9999996699 = 99999834000043899999
For factor length 11, 99999943851 * 99999996349 = 9999994020000204999999
For factor length 12, 999999000001 * 999999999999 = 999999000000000000999999
```

### Faster version

Translation of: Python
```""" taken from https://leetcode.com/problems/largest-palindrome-product/discuss/150954/Fast-algorithm-by-constrains-on-tail-digits """

const T = [Set([(0, 0)])]

function double(it)
arr = empty(it)
for p in it
push!(arr, p, reverse(p))
end
return arr
end

""" Construct a pair of n-digit numbers such that their product ends with 99...9 pattern """
function tails(n)
if length(T) <= n
l = Set()
for i in 0:9, j in i:9
I = i * 10^(n-1)
J = j * 10^(n-1)
it = collect(tails(n - 1))
I != J && (it = double(it))
for (t1, t2) in it
if ((I + t1) * (J + t2) + 1) % 10^n == 0
push!(l, (I + t1, J + t2))
end
end
end
push!(T, l)
end
return T[n + 1]
end

""" find the largest palindrome that is a product of n-digit numbers """
function largestpalindrome(n)
m, tail = 0, n ÷ 2
for L in 1 : 9 * 10^(head-1)
# Consider small shell (up-L)^2 < (up-i)*(up-j) <= (up-L)^2, 1<=i<=L<=j
m, sol = 0, (0, 0)
for i in 1:L
lo = max(Int128(i), Int128(up - (up - L + 1)^2 ÷ (up - i)) + 1)
hi = Int128(up - (up - L)^2 ÷ (up - i))
for j in lo:hi
I = (up - i) * 10^tail
J = (up - j) * 10^tail
it = collect(tails(tail))
I != J && (it = double(it))
for (t1, t2) in it
val = (I + t1) * (J + t2)
s = string(val)
if s == reverse(s) && val > m
sol = (I + t1, J + t2)
m = val
end
end
end
end
if m > 0
println(lpad(n, 2), "    ", lpad(m % 1337, 4), " \$sol \$(sol * sol)")
return m % 1337
end
end
return 0
end

@time for k in 1:16
largestpalindrome(k)
end
```
Output:
``` 1       9 (9, 1) 9
2     987 (91, 99) 9009
3     123 (993, 913) 906609
4     597 (9901, 9999) 99000099
5     677 (99979, 99681) 9966006699
6    1218 (999001, 999999) 999000000999
7     877 (9998017, 9997647) 99956644665999
8     475 (99990001, 99999999) 9999000000009999
9    1226 (999980347, 999920317) 999900665566009999
10     875 (9999986701, 9999996699) 99999834000043899999
11     108 (99999943851, 99999996349) 9999994020000204999999
12     378 (999999000001, 999999999999) 999999000000000000999999
13    1097 (9999999993349, 9999996340851) 99999963342000024336999999
14     959 (99999990000001, 99999999999999) 9999999000000000000009999999
15     465 (999999998341069, 999999975838971) 999999974180040040081479999999
16      51 (9999999900000001, 9999999999999999) 99999999000000000000000099999999
62.575515 seconds (241.50 M allocations: 16.491 GiB, 25.20% gc time, 0.07% compilation time)
```

## Mathematica / Wolfram Language

```palindromeQ[n_] := (* faster than built in test PalindromeQ *)
Block[{digits = IntegerDigits@n}, digits == Reverse[digits]]

nextPair[n_] := (* outputs next pair of candidate divisors *)
Block[{next =
NestWhile[# - 11 &, n, ! MemberQ[{1, 3, 7, 9}, Mod[#, 10]] &],
len = Last@RealDigits@n},
{next, 10^len - Switch[Mod[next, 10], 1, 1, 3, 7, 7, 3, 9, 9]}]

search[n_] :=
Block[{resetLimit = 10^(n - Floor[n/2]) (10^Floor[n/2] - 1), cands},
cands =
Partition[
Flatten[
Reap[
NestWhile[(If[palindromeQ[Times @@ #], Sow[#]];
If[Last@# < resetLimit,
nextPair[First@# - 11], # - {0, 10}]) &,
nextPair@If[EvenQ@n, 10^n - 1, 10^n - 21],
First@# > resetLimit &]]], 2];
Flatten@cands[[Ordering[Times @@@ cands, -1]]]]

Grid[Join[{{"factors", "largest palindrome"}}, {#, Times @@ #} & /@
Table[search[n], {n, 2, 7}]], Alignment -> {Left, Baseline}]
```
Output:
```
factors			largest palindrome
{99,91}			9009
{913,993}       	906609
{9999,9901}     	99000099
{99979,99681}   	9966006699
{999999,999001} 	999000000999
{9997647,9998017}	99956644665999

```

## Paper & Pencil

```find two 3-digit factors, that when multiplied together, yield the highest 6-digit palindrome.

lowest possible 6 digit palindrome starting with 9 is 900009
floor(sqrt(900009)) = 948
one factor must be an odd multiple of 11
floor(948 / 11) = 86
it must not be even or a multiple of 5, so use 83
83 * 11 = 913 <- this is the lowest possible first factor
the last digit of the second factor must multiply with the last digit of the first factor to get 9
the highest suitable second factor (for 913) is 993
913 x 993 = 906609, a palindrome, now check suitable higher first factors
913 + 22 = 935, an unsuitable multiple of 5, so skip it and use 913 + 44 = 957
957 x 997 = ‭954129‬, not a palindrome, so continue (just subtract 9570)
957 x 987 = 944559‬‬, not a palindrome, so continue
957 x 977 = ‭934989‬, not a palindrome, so continue
957 x 967 = ‭925429‬, not a palindrome, so continue
957 x 957 = ‭915849‬, not a palindrome, so continue
957 x 947 = ‭906279‬, not a palindrome, and less than the best found so far, so stop and
continue to the next suitible first number, 957 + 22 = 979
979 x 991 = 970189‬‬, not a palindrome, so continue (just subtract 9790)
979 x 981 = 960399‬, not a palindrome, so continue
979 x 971 = 950609‬, not a palindrome, so continue
979 x 961 = 940819‬, not a palindrome, so continue
979 x 951 = 931029‬, not a palindrome, so continue
979 x 941 = 921239‬, not a palindrome, so continue
979 x 931 = 911449‬, not a palindrome, so continue
979 x 921 = 901659‬, not a palindrome, and less than the best found so far, so stop
done because 979 + 22 = 1001
```

## Perl

Library: ntheory
```use strict;
use warnings;
use feature 'say';
use ntheory 'divisors';

for my \$l (2..7) {
LOOP:
for my \$p (reverse map { \$_ . reverse \$_ } 10**(\$l-1) .. 10**\$l - 1)  {
my @f = reverse grep { length == \$l } divisors \$p;
next unless @f >= 2 and \$p == \$f * \$f;
say "Largest palindromic product of two @{[\$l]}-digit integers: \$f × \$f = \$p" and last LOOP;
}
}
```
Output:
```Largest palindromic product of two 2-digit integers: 91 × 99 = 9009
Largest palindromic product of two 3-digit integers: 913 × 993 = 906609
Largest palindromic product of two 4-digit integers: 9901 × 9999 = 99000099
Largest palindromic product of two 5-digit integers: 99681 × 99979 = 9966006699
Largest palindromic product of two 6-digit integers: 999001 × 999999 = 999000000999
Largest palindromic product of two 7-digit integers: 9997647 × 9998017 = 99956644665999```

## Phix

Library: Phix/online

Translated from python by Lucy_Hedgehog as found on page 5 of the project euler discussion page (dated 25 Sep 2011), and further optimised as per the C# comments (on this very rosettacode page). You can run this online here.

```-- demo\rosetta\Largest_palindrome_product.exw
with javascript_semantics
requires("1.0.1") -- (mpz_fdiv_qr(), mpz_si_sub() added to mpfr.js, mpz_mod_ui(), mpz_fdiv_q_ui(), mpz_fdiv_r(), mpz_fdiv_ui() fixed)
include mpfr.e

function ispalindrome(mpz x)
string s = mpz_get_str(x)
return s == reverse(s)
end function

function inverse(mpz x, integer m)
-- Compute the modular inverse of x modulo power(10,m).
-- Return -1 if the inverse does not exist.
-- This function uses Hensel lifting.
integer a = {-1, 1, -1, 7, -1, -1, -1, 3, -1, 9}[mpz_fdiv_ui(x,10)+1]
if a!=-1 then
mpz ax = mpz_init()
while true do
mpz_mul_si(ax,x,a)
{} = mpz_mod_ui(ax,ax,m)
if mpz_cmp_si(ax,1)==0 then exit end if
mpz_si_sub(ax,2,ax)
mpz_mul_si(ax,ax,a)
a = mpz_fdiv_q_ui(ax,ax,m)
end while
end if
return a
end function

function pal2(integer n)
assert(n>1)

mpz {best,factor,y,r} = mpz_inits(4)
if even(n) then
// (as per the C# comments)
mpz_ui_pow_ui(factor,10,n/2)
mpz_sub_si(factor,factor,1)
mpz_ui_pow_ui(best,10,n/2*3)
mpz_mul(best,best,factor)
assert(ispalindrome(best))
mpz_ui_pow_ui(factor,10,n)
mpz_sub_si(factor,factor,1)
assert(ispalindrome(factor))
else
// Get a lower bound:
integer k = floor(n/2)
mpz {maxf,maxf11,minf,x,t,maxy,p} = mpz_inits(7)
while true do
mpz_ui_pow_ui(maxf,10,n)
mpz_sub_si(maxf,maxf,1)
mpz_sub_si(maxf11,maxf,11)
{} = mpz_fdiv_q_ui(maxf11,maxf11,22)
mpz_mul_si(maxf11,maxf11,22)
mpz_ui_pow_ui(minf,10,n-k)
mpz_sub(minf,maxf,minf)
mpz_mul(best,minf,minf)
mpz_set_si(factor,0)
// This palindrome starts with k 9's.
// Hence the largest palindrom must also start with k 9's and
// therefore end with k 9's.
// Thus, if p = x * y is the solution then
// x * y + 1 is divisible by m.
integer m = power(10,k) -- (should not exceed 1e8)
mpz_set(x,maxf11)
while mpz_cmp_si(x,1)>=0 do
mpz_mul(t,x,maxf)
if mpz_cmp(t,best)=-1 then exit end if
integer ry = inverse(x, m)
if ry!=-1 then
mpz_mul(p,maxy,x)
while mpz_cmp(p,best)>0 do
if ispalindrome(p) then
mpz_set(best,p)
mpz_set(factor,x)
end if
mpz_mul_si(t,x,m)
mpz_sub(p,p,t)
end while
end if
mpz_sub_si(x,x,22)
end while
if mpz_cmp_si(factor,0)!=0 then exit end if
k -= 1
end while
end if
mpz_fdiv_qr(y,r,best,factor)
assert(mpz_cmp_si(r,0)=0)
return {best, factor, y}
end function

constant fmt = "Largest palindromic product of two %d-digit integers: %s = %s x %s (%s)\n"
for n=2 to 12 do
atom t1 = time()
mpz {p,x,y} = pal2(n)
string sp = mpz_get_str(p),
sx = mpz_get_str(x),
sy = mpz_get_str(y),
e = elapsed(time()-t1)
printf(1,fmt,{n,sp,sx,sy,e})
end for
```
Output:
```Largest palindromic product of two 2-digit integers: 9009 = 99 x 91 (0s)
Largest palindromic product of two 3-digit integers: 906609 = 913 x 993 (0s)
Largest palindromic product of two 4-digit integers: 99000099 = 9999 x 9901 (0s)
Largest palindromic product of two 5-digit integers: 9966006699 = 99979 x 99681 (0s)
Largest palindromic product of two 6-digit integers: 999000000999 = 999999 x 999001 (0s)
Largest palindromic product of two 7-digit integers: 99956644665999 = 9997647 x 9998017 (0.0s)
Largest palindromic product of two 8-digit integers: 9999000000009999 = 99999999 x 99990001 (0s)
Largest palindromic product of two 9-digit integers: 999900665566009999 = 999920317 x 999980347 (0.8s)
Largest palindromic product of two 10-digit integers: 99999000000000099999 = 9999999999 x 9999900001 (0s)
Largest palindromic product of two 11-digit integers: 9999994020000204999999 = 99999996349 x 99999943851 (0.1s)
Largest palindromic product of two 12-digit integers: 999999000000000000999999 = 999999999999 x 999999000001 (0s)
```

After that it starts to struggle a bit:

```Largest palindromic product of two 13-digit integers: 99999963342000024336999999 = 9999996340851 x 9999999993349 (40.4s)
Largest palindromic product of two 14-digit integers: 9999999000000000000009999999 = 99999999999999 x 99999990000001 (0s)
Largest palindromic product of two 15-digit integers: 999999974180040040081479999999 = 999999998341069 x 999999975838971 (1 minute and 12s)
Largest palindromic product of two 16-digit integers: 99999999000000000000000099999999 = 9999999999999999 x 9999999900000001 (0s)
```

## Python

Original author credit to user peijunz at Leetcode.

```""" taken from https://leetcode.com/problems/largest-palindrome-product/discuss/150954/Fast-algorithm-by-constrains-on-tail-digits """

T=[set([(0, 0)])]

def double(it):
for a, b in it:
yield a, b
yield b, a

def tails(n):
'''Construct pair of n-digit numbers that their product ends with 99...9 pattern'''
if len(T)<=n:
l = set()
for i in range(10):
for j in range(i, 10):
I = i*10**(n-1)
J = j*10**(n-1)
it = tails(n-1)
if I!=J: it = double(it)
for t1, t2 in it:
if ((I+t1)*(J+t2)+1)%10**n == 0:
T.append(l)
return T[n]

def largestPalindrome(n):
""" find largest palindrome that is a product of two n-digit numbers """
m, tail = 0, n // 2
# Consider small shell (up-L)^2 < (up-i)*(up-j) <= (up-L)^2, 1<=i<=L<=j
m = 0
sol = None
for i in range(1, L + 1):
lo = max(i, int(up - (up - L + 1)**2 / (up - i)) + 1)
hi = int(up - (up - L)**2 / (up - i))
for j in range(lo, hi + 1):
I = (up-i) * 10**tail
J = (up-j) * 10**tail
it = tails(tail)
if I!=J: it = double(it)
for t1, t2 in it:
val = (I + t1)*(J + t2)
s = str(val)
if s == s[::-1] and val>m:
sol = (I + t1, J + t2)
m = val

if m:
print("{:2d}\t{:4d}".format(n, m % 1337), sol, sol * sol)
return m % 1337
return 0

if __name__ == "__main__":
for k in range(1, 14):
largestPalindrome(k)
```
Output:
``` 1	   9 (9, 1) 9
2	 987 (91, 99) 9009
3	 123 (993, 913) 906609
4	 597 (9901, 9999) 99000099
5	 677 (99979, 99681) 9966006699
6	1218 (999001, 999999) 999000000999
7	 877 (9998017, 9997647) 99956644665999
8	 475 (99990001, 99999999) 9999000000009999
9	1226 (999980347, 999920317) 999900665566009999
10	 875 (9999986701, 9999996699) 99999834000043899999
11	 108 (99999943851, 99999996349) 9999994020000204999999
12	 378 (999999000001, 999999999999) 999999000000000000999999
13	1097 (9999999993349, 9999996340851) 99999963342000024336999999
```

## Quackery

The largest product of two 3 digit numbers is 999*999 = 998001. The smallest product of two 3 digit numbers is 100*100 = 10000. Therefore we need only consider 6 and 5 digit palindromic numbers.

Considering 6 digit palindromic numbers:

The largest is 999999. The smallest is 100001. These can be costructed from the numbers 100 to 999. Therefore there are 899 6 digit palindromic numbers.

The same applies to 5 digit palindromic numbers: The largest is 99999. The smallest is 10001. These can be costructed from the numbers 100 to 999. Therefore there are 899 5 digit palindromic numbers.

Method:

• Construct the 6 digit palindromic numbers in reverse numerical order.
• Test each one to see if it is divisible by a 3 digit number with a 3 digit result, starting with 999.
• If it is, the solution has been found.
• If no solution found, consider 5 digit palindromes.

A six digit solution was found, and as it was found virtually instantaneously I did not feel that any optimisations were necessary.

I went on to find the largest five digit soultion, even though the task did not call for it, as it was a trivial exercise.

```  [ [] swap
[ 10 /mod
rot join swap
dup 0 = until ]
drop ]              is ->digits (   n --> [ )

witheach
[ swap 10 * + ] ] is ->number (   [ --> n )

[ ->digits
dup reverse join
->number ]          is evenpal  (   n --> n )

[ ->digits
dup reverse
->number ]          is oddpal   (   n --> n )

[ 2dup mod 0 != iff
[ 2drop false ]
done
/ 100 1000 within ] is solution ( n n --> b )

false
899 times
[ i 100 + evenpal
899 times
[ dup i 100 +
solution if
[ dip not
conclude ] ]
over iff
[ nip conclude ]
else drop ]
dup iff
[ say "Six digit solution found: " echo ]
else
[ drop say "No six digit solution found." ]
cr
false
899 times
[ i 100 + oddpal
899 times
[ dup i 100 +
solution if
[ dip not
conclude ] ]
over iff
[ nip conclude ]
else drop ]
dup iff
[ say "Five digit solution found: " echo ]
else
[ drop say "No five digit solution found." ]```
Output:
```Six digit solution found: 906609
Five digit solution found: 99899
```

## Raku

```use Inline::Perl5;
my \$p5 = Inline::Perl5.new();
\$p5.use: 'ntheory';
my &divisors = \$p5.run('sub { ntheory::divisors \$_ }');

.say for (2..12).map: {.&lpp};

multi lpp (\$oom where {!(\$_ +& 1)}) { # even number of multiplicand digits
my \$f = +(9 x \$oom);
my \$o = \$oom / 2;
my \$pal = +(9 x \$o ~ 0 x \$oom ~ 9 x \$o);
sprintf "Largest palindromic product of two %2d-digit integers: %d × %d = %d", \$oom, \$pal div \$f, \$f, \$pal
}

multi lpp (\$oom where {\$_ +& 1}) { # odd number of multiplicand digits
my \$p;
(+(1 ~ (0 x (\$oom - 1))) .. +(9 ~ (9 x (\$oom - 1)))).reverse.map({ +(\$_ ~ .flip) }).map: -> \$pal {
for my @factors = divisors("\$pal")».Int.grep({ .chars == \$oom }).sort( -* ) {
next unless \$pal div \$_ ∈ @factors;
\$p = sprintf("Largest palindromic product of two %2d-digit integers: %d × %d = %d", \$oom, \$pal div \$_, \$_, \$pal);
last;
}
last if \$p;
}
\$p
}
```
```Largest palindromic product of two  2-digit integers: 91 × 99 = 9009
Largest palindromic product of two  3-digit integers: 913 × 993 = 906609
Largest palindromic product of two  4-digit integers: 9901 × 9999 = 99000099
Largest palindromic product of two  5-digit integers: 99681 × 99979 = 9966006699
Largest palindromic product of two  6-digit integers: 999001 × 999999 = 999000000999
Largest palindromic product of two  7-digit integers: 9997647 × 9998017 = 99956644665999
Largest palindromic product of two  8-digit integers: 99990001 × 99999999 = 9999000000009999
Largest palindromic product of two  9-digit integers: 999920317 × 999980347 = 999900665566009999
Largest palindromic product of two 10-digit integers: 9999900001 × 9999999999 = 99999000000000099999
Largest palindromic product of two 11-digit integers: 99999943851 × 99999996349 = 9999994020000204999999
Largest palindromic product of two 12-digit integers: 999999000001 × 999999999999 = 999999000000000000999999```

## Ring

```? "working..."

prod = 1
bestProd = 0
// maximum 3 digit number
max = 999
// both factors must be >100 for a 6 digit product
limitStart = 101
// one factor must be divisible by 11
limitEnd = 11 * floor(max / 11)
second = limitStart
iters = 0

// loop from hi to low to find the best result in the fewest steps
for n = limitEnd to limitStart step -11
// with n falling, the lower limit of m can rise with
// the best-found-so-far second number. Doing this
// lowers the iteration count by a lot.
for m = max to second step -2
prod = n * m
if isPal(prod)
iters++
// exit when the product stops increasing
if bestProd > prod
exit
ok
// maintain the best-found-so-far result
first = n
second = m
bestProd = prod
ok
next
next

put "The largest palindrome is: "
? "" + bestProd + " = " + first + " * " + second
? "Found in " + iters + " iterations"
put "done..."

func isPal n
x = string(n)
l = len(x) + 1
i = 0
while i < l
if x[i++] != x[l--]
return false
ok
end
return true```
Output:
```working...
The largest palindrome is: 906609 = 913 * 993
Found in 6 iterations
done...```

## RPL

```≪ "" OVER SIZE 1 FOR j
OVER j DUP SUB +
-1 STEP NIP
≫ 'REVSTR' STO

≪ 1 CF
DO
1 -
DUP →STR DUP REVSTR + STR→
DUP DIVIS SORT
1 OVER SIZE FOR j
DUP j GET
DUP XPON 2
CASE
DUP2 < THEN 3 DROPN END
> THEN DROP DUP SIZE 'j' STO END
PICK3 SWAP /
IF XPON 2 == THEN 1 SF END
END
NEXT DROP2
UNTIL 1 FS? END
→STR DUP REVSTR + STR→
≫ 'P004' STO
```
```1000 P004
```
Output:
```1: 906609
```

## Sidef

```func largest_palindrome_product (n) {

for k in ((10**n - 1) `downto` 10**(n-1)) {
var t = Num("#{k}#{Str(k).flip}")

t.divisors.each {|d|
if ((d.len == n) && ((t/d).len == n)) {
return (d, t/d)
}
}
}
}

for n in (2..9) {
var (a,b) = largest_palindrome_product(n)
say "Largest palindromic product of two #{n}-digit integers: #{a} * #{b} = #{a*b}"
}
```
Output:
```Largest palindromic product of two 2-digit integers: 91 * 99 = 9009
Largest palindromic product of two 3-digit integers: 913 * 993 = 906609
Largest palindromic product of two 4-digit integers: 9901 * 9999 = 99000099
Largest palindromic product of two 5-digit integers: 99681 * 99979 = 9966006699
Largest palindromic product of two 6-digit integers: 999001 * 999999 = 999000000999
Largest palindromic product of two 7-digit integers: 9997647 * 9998017 = 99956644665999
Largest palindromic product of two 8-digit integers: 99990001 * 99999999 = 9999000000009999
Largest palindromic product of two 9-digit integers: 999920317 * 999980347 = 999900665566009999
```

## Wren

The approach here is to manufacture palindromic numbers of length 2n in decreasing order and then see if they're products of two n-digit numbers.

```var reverse = Fn.new { |n|
var r = 0
while (n > 0) {
r = n%10 + r*10
n = (n/10).floor
}
return r
}

var pow = 10
for (n in 2..7) {
var low = pow * 9
pow = pow * 10
var high = pow - 1
System.write("Largest palindromic product of two %(n)-digit integers: ")
var nextN = false
for (i in high..low) {
var j = reverse.call(i)
var p = i * pow + j
// k can't be even nor end in 5 to produce a product ending in 9
var k = high
while (k > low) {
if (k % 10 != 5) {
var l = p / k
if (l > high) break
if (p % k == 0) {
System.print("%(k) x %(l) = %(p)")
nextN = true
break
}
}
k = k - 2
}
if (nextN) break
}
}```
Output:
```Largest palindromic product of two 2-digit integers: 99 x 91 = 9009
Largest palindromic product of two 3-digit integers: 993 x 913 = 906609
Largest palindromic product of two 4-digit integers: 9999 x 9901 = 99000099
Largest palindromic product of two 5-digit integers: 99979 x 99681 = 9966006699
Largest palindromic product of two 6-digit integers: 999999 x 999001 = 999000000999
Largest palindromic product of two 7-digit integers: 9998017 x 9997647 = 99956644665999
```

## XPL0

```func Rev(A);    \Reverse digits
int A, B;
[B:= 0;
repeat  A:= A/10;
B:= B*10 + rem(0);
until   A = 0;
return B;
];

int Max, M, N, Prod;
[Max:= 0;
for M:= 100 to 999 do
for N:= M to 999 do
[Prod:= M*N;
if Prod/1000 = Rev(rem(0)) then
if Prod > Max then Max:= Prod;
];
IntOut(0, Max);
]```
Output:
```906609
```