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Palindromic Products of Integers and their Reversals
Sum of First Numbertypes Sequence Products Pythagorean Triples Palindromes in other Bases Palindromes in Concatenations Various Palindromic Sums

Palindromic numbers are numbers which read the same from

  left to right (forwards) as from the right to left (backwards)

Here are a few random examples : 505, 137731, 100246642001

Palindromic Products of Non palindromic Integers and their Reversals are defined and calculated by this extraordinary intricate
and excruciatingly complex formula. So, this line is for experts only

base  x  esab

Here is a gem from Carlos Rivera's puzzle website ( Source Puzzle 52 ).

Palindromic sum of two consecutive primes 32213 & 32233 equal to the sum of their reversibles

32213 + 32233 =  64446  = 33223 + 31223

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A 3-digit number trick.

Everybody must have heard of the following number trick that involves 'reverse & add'-ing
the digits of a random 3-digit number. If not here your chance to catch up.

I'm grateful to Mitch Beck for making me aware of this property of 1089, a truly funny number.

Take any three integers from zero to nine, then subtract its reversal.
Then, if the difference is positive, add its reversal.
If the difference is negative, then subtract its reversal.
NO MATTER WHAT 3-DIGIT INTEGERS YOU BEGIN WITH, THE FINAL ANSWER IS ALWAYS 1089 !

            856        159        872
          - 658      - 951      - 278
         ------     ------     ------
            198      - 792        594
          + 891      - 297      + 495
         ------     ------     ------
           1089      -1089       1089

“Curious and Interesting Numbers”, by David Wells, page 163, about 1089 :

If a 3-digit number is reversed and the result subtracted,

and that answer added to its reversal, the answer is always 1089:

623 - 326 = 297 and 297 + 792 = 1089.

“Recreations in the Theory of Numbers”, by Albert H. Beiler, page 63, says :

A common trick with an almost infinite number of variations is to

have someone write a three-digit number, then write the number

with the digits in reverse order, subtract the smaller from the larger,

reverse the digits again and add this time; you will then be able to

give the answer without having received any information about the

number. The answer is 1089. Suppose any number, say 173, were

taken. Subtract from 371, leaving 198, reverse, giving 891, and the

sum of the last two numbers is 1089.

On the following page 64 Beiler gives a mathematical proof why this operation always works.
It might be worthwhile to order this book as it is almost a bible for number theory enthousiasts.

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[ September 10, 2002 ]
Sathiya Subramanian (email) from Redwood City, California
struggled with what he thought is a counterexample of the above 3-digit number trick.
He used integer 758 to start with ¬

   758
– 857
--------
– 99
– 99
--------
– 189 ??

The trick here is to keep using three digits even if a leading zero
appears in the result ! Sathiya's example thus becomes ¬

   758
– 857
--------
– 099
– 990
--------
– 1089 !!

A very sharp observation nevertheless. Thanks!

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PS. 1089 is also known for another conjecture.
When an integer and its reversal are unequal, their product is never a square except when both are squares.

1089 = 33² and 9801 = 99²

We see that 1089 x 9801 = 10673289 which is 3267².

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Some observations about Palindromic Products of Integers & their Reversals.
[See table at the end of this page]

The basenumbers are made up of a combination of only three digits namely 0, 1 and 2 !
Although it is not the case, one could mistakenly believe that we are dealing with numbers in base 3 !!
Is someone prepared to find out why that is ? A mathematical proof will be much appreciated.

[ August 9, 2000 ]
Henry Bottomley (email) cannot quite give a proof yet, but his explanation is no less interesting.

If the sum of the squares of the digits of n is less than 10 and n is not
divisible by 10, then the product of n and the reversal of n is a palindrome.
[Indeed extending this to any base k: if the sum of the squares of the base k
digits of n is less than k and n is not divisible by k, then the product of n
and the base k reversal of n is a base k palindrome.]

The reason for this is: 
(i) any (perhaps partial) sum of pairwise products of digits of n is less than
or equal to the sum of the square of the digits of n, which in turn is less
than 10 [or k];
(ii) therefore there are no carries when calculating the digits of the product
of n and its reversal;
(iii) since multiplication and addition are commutative, this means that each
digit of the product is equal to the digit an equal distance the other side
of the middle of the product;
(iv) and since n is not a multiple of 10 [or k] there are no difficulties with
final zeros in the product.

To take your example, the digits of base*esab are (before any carries):
b*e; b*s+a*e; b*a+a*s+s*e; b*b+a*a+s*s+e*e; a*b+s*a+e*s; s*b+e*a; e*b
All of these are less or equal to b*b+a*a+s*s+e*e which is less than 10
[or k], so there are no carries.
So this appears to be a palindrome.
Since e is not zero (and neither is b), e*b is not zero, and so indeed this
is a palindrome.

If the condition is not met, palindromes are (probably) impossible, because
of the carries.

So the possibilities for digits are: 

• 3;
• two 2s, with one or zero 1s, and as many 0s as required (but not as a final digit);
• one 2, with from zero through to five 1s, and as many 0s as required (but not as a final digit);
• from two through to nine 1s, and as many 0s as required (but not as a final digit);
• 1;
• 0 (as a special case).

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Explorations into the reversal world

The table lists all the palindromes only up to length 9. Larger ones are very easy to find.
Per given length their total number grows exponentially.

     Length of      Total genuine 
     basenumber     n x n_reversed
          1               0
          2               1
          3               3
          4              10
          5              19
          6              44
          7               ?

That doesn't mean that we can't find some larger palindromes with beautiful patterns or special properties.
On the contrary. What about repunital palindromes except for the middle digit :

1011 x 1101 = 1113111

1010011 x 1100101 = 1111114111111

I'd like to name these palindromes wingnumbers, for obvious reason.
Here is another one from More Palindromic Products of Integer Sequences :

1051 x 1061 = 1115111

This same palindromic wingnumber has also another strange property (See Sloane's A032735).
It cannot be prefixed or followed by any digit from (1,2,3,4,5,6,7,8,9) to form a prime.

211151117, 511151115, 911151116, etc. are all composite !

And this wingnumber comes from the Nine Digits Page 2 :

1⁸ + 2⁵ + 3⁹ + 4² + 5³ + 6¹ + 7⁷ + 8⁶ + 9⁴ = 1112111

Fascinating digit-swapping occurs many times with e.g. these two reversals :

    1 0 2 1 x 1 2 0 1 = 1 2 2 6 2 2 1
         X     X         X         X
    1 0 1 2 x 2 1 0 1 = 2 1 2 6 2 1 2

Here is a finite pattern starting with the very first one of the table :

2 x 2 = 4
12 x 21 = 252
112 x 211 = 23632
1112 x 2111 = 2347432
11112 x 21111 = 234585432
111112 x 211111 = 23456965432

Infinite patterns are abundant here. But this one has something special ! Read on.

1121 x 1211 = 1357531

1010201 x 1020101 = 1030507050301

1001002001 x 1002001001 = 1003005007005003001

1000100020001 x 1000200010001 = 1000300050007000500030001

And the following phenomenon totally surprised me !
All the first terms of the above pattern when added together with the square of their previous (less one) numbers
( see Quasi-Over-Squares ) also produce palindromes :

1120² + 1121 = 1255521

1010200² + 1010201 = 1020505050201

1001002000² + 1001002001 = 1002005005005002001

1000100020000² + 1000100020001 = 1000200050005000500020001

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David Wilson found out an interesting palindrome-related fact
Article emailed at [ Sun, 08 Mar 1998 ]

The number 1089 (the square of palindrome 33) is interesting
because it reverses when multiplied by 9: 9 * 1089 = 9801.
Well, it turns out that multiplication by 9 reverses a positive
integer if and only if that integer is of the form 99*p, where
p is a palindrome consisting of only the digits 0 and 1, in
which every run of 0's or 1's is at least two digits long.

For instance, the palindrome 11000111100011 fits the bill for p.
Then 99 * p = 1089010998901089, and we see that

    9 * 1089010998901089 = 9801098990109801.

Similarly, the set of numbers which reverses when multiplied
by 4 are those of the form 198 * p, where p is of the same
above-described palindromic form.

Finally, it is obvious that the palindromes themselves are
reversed when multiplied by 1.  This exhausts the posibilities;
in base 10, the only multipliers that can reverse a positive
integer are 1, 4, or 9.

There is some evidence that similar types of things happen
in other bases.  For instance, in base 5, a number of the
form 13 * p, where p is any palindrome consisting of 0's and 1's
is reversed when multiplied by 2.

Eric Weisstein added an interesting article to his Math Encyclopedia about Reversals which are integral multiples of themselves.

2178 x 4 = 8712

219...978 x 4 = 879...912

I cannot withhold the following equation althought the numbers are not palindromic :
[ Source “Curious and Interesting Numbers” by David Wells, page 158 ]

651 x 156 = 372 x 273

Neil Sloane's “Integer Sequences” Encyclopedia can be consulted online : Neil Sloane's Integer Sequences Some prime entries about the 'reversal' subject are e.g. : %N Primes whose reversal is prime. under A007500. %N Emirps (primes whose reversal is a different prime). under A006567. Some palindromic entries about the 'reversal' subject are e.g. : %N Number of iterations of Reverse and Add which lead to a palindrome. under A016016. %N Least integer requiring n iterations of Reverse and Add to reach a palindrome. under A015994. %N 1 iteration of Reverse and Add is needed to reach a palindrome. under A015976. %N 2 iterations of Reverse and Add are needed to reach a palindrome. under A015977. %N 3 iterations of Reverse and Add are needed to reach a palindrome. under A015979. %N 4 iterations of Reverse and Add are needed to reach a palindrome. under A015980. %N 5 iterations of Reverse and Add are needed to reach a palindrome. under A015982. %N 6 iterations of Reverse and Add are needed to reach a palindrome. under A015984. %N 7 iterations of Reverse and Add are needed to reach a palindrome. under A015986. %N 8 iterations of Reverse and Add are needed to reach a palindrome. under A015988. %N 9 iterations of Reverse and Add are needed to reach a palindrome. under A015990. %N 10 iterations of Reverse and Add are needed to reach a palindrome. under A015991. %N 11 iterations of Reverse and Add are needed to reach a palindrome. under A015992. %N 12 iterations of Reverse and Add are needed to reach a palindrome. under A015993. %N Integers n such that the 'Reverse and Add!' trajectory of n (presumably) does not reach        a palindrome and does not join the trajectory of any m < n. under A063048. General reversal sequences : %N a(n) * a(n)_reversed is a palindrome (and a(n) is not palindromic). under A048344. %N Total number of distinct palindromes of form 'n-digit number' *        'n-digit number_reversed' (n-digit number not palindromic). under A048343. %N Numbers n such that n*R(n) is a palindrome, where R(n) = the digit reversal. under A062936. Click here to view some of the author's [P. De Geest] entries to the table. Click here to view some entries to the table about palindromes.

Index Nr	Info	Basenumber	Length
	Palindromic Product of Integers & their Reversals		Length
	Integer[Normal]  x  Integer[Reversed]
Halted with length 9 (See also A048343 and A048344)
33	20.012 x 21.002		5
	420.292.024		9
32	12.002 x 20.021		5
	240.292.042		9
31	11.121 x 12.111		5
	134.686.431		9
30	11.112 x 21.111		5
	234.585.432		9
29	11.102 x 20.111		5
	223.272.322		9
28	11.021 x 12.011		5
	132.373.231		9
27	11.012 x 21.011		5
	231.373.132		9
26	11.002 x 20.011		5
	220.161.022		9
25	10.211 x 11.201		5
	114.373.411		9
24	10.202 x 20.201		5
	206.090.602		9
23	10.121 x 12.101		5
	122.474.221		9
22	10.112 x 21.101		5
	213.373.312		9
21	10.111 x 11.101		5
	112.242.211		9
20	10.102 x 20.101		5
	203.060.302		9
19	10.022 x 22.001		5
	220.494.022		9
18	10.021 x 12.001		5
	120.262.021		9
17	10.012 x 21.001		5
	210.262.012		9
16	10.011 x 11.001		5
	110.131.011		9
15	10.002 x 20.001		5
	200.050.002		9
14	2.012 x 2.102		4
	4.229.224		7
13	1.202 x 2.021		4
	2.429.242		7
12	1.121 x 1.211		4
	1.357.531		7
11	1.112 x 2.111		4
	2.347.432		7
10	1.102 x 2.011		4
	2.216.122		7
9	1.022 x 2.201		4
	2.249.422		7
8	1.021 x 1.201		4
	1.226.221		7
7	1.012 x 2.101		4
	2.126.212		7
6	1.011 x 1.101		4
	1.113.111		7
5	1.002 x 2.001		4
	2.005.002		7
4	122 x 221		3
	26.962		5
3	112 x 211		3
	23.632		5
2	102 x 201		3
	20.502		5
1	12 x 21		2
	252		3

David W. Wilson (email) found out an interesting palindrome-related fact - go to topic

Henry Bottomley (email) gives an explanation why the basenumbers are composed
only of digits 0, 1 and 2 - go to topic

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( © All rights reserved ) - Last modified : July 3, 2023.

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Patrick De Geest - Belgium - Short Bio - Some Pictures
E-mail address : pdg@worldofnumbers.com

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