Introduction

The numbers, which gives a perfect square on adding as well as subtracting its reverse are rare and hence termed as Rare Numbers[1][2].

If R is a positive integer and R1 is the integer obtained from R by writing its decimal digits in reverse order, then if R + R1 and R - R1 both are perfect square then R is termed as Rare Number.

So for R to be a Rare Number we must have

R + R1 = X² and R - R1 = Y²


For example:

For R=65, R1=56

R+R1 = 65+56 = 121 = 11²

R-R1 = 65 - 56 = 9 = 3²

So 65 is a Rare Number [3].

Similarly for R=621770, R1=77126

R+R1 = 621770 + 77126 = 698896 = 836²

R-R1 = 621770 - 77126 = 544644 = 738²

So 621770 is a Rare Number [7].

Other Example of Rare numbers are 281089082, 2022652202, 2042832002, 868591084757, 872546974178, 872568754178, 6979302951885, 20313693904202, 20313839704202, 20331657922202, 20331875722202, 20333875702202, 40313893704200, 40351893720200, 200142385731002, 204238494066002......(Sequence A035519)

   Palindromic Rare Numbers

The examples given above are non-palindromic. In case palindromic numbers are also considered then it is noticed that there are infinite number of palindromic Rare numbers [4].

For example : R=242 gives R+R1 = 484 = 22² and R-R1=0.

Since the numbers in the series 242,20402,2004002, .... are all palindromic Rare numbers, it is obvious that there are infinite number of palindromic Rare numbers.

The other example of infinite series of palindromic Rare numbers are 24642, 204060402, 2004006004002,... and 2468642, 2040608060402, 2004006008006004002,....

Unlike palindromic Rare numbers which are infinite, it still remains to find whether non-palindromic Rare numbers are finite or infinite.

   Properties of Rare Numbers

Some of the properties of Rare Numbers are given below.

Consider a Rare number ABCD...MNPQ with any number of digits, then

(i) The value of A can only be 2, 4, 6 or 8 i.e. Rare number can never start with an odd digit.


(ii) If A = 2 then Q = 2 and B = P.

If A = 4 then Q = 0 and B - P = positive or negative Even digit i.e. -8, -6, -4, -2, 0, 2, 4, 6, 8.
If A = 6 then Q = 0 or 5 and B - P = positive or negative Odd digit i.e. -9, -7, -5, -3, -1, 1, 3, 5, 7, 9.
If A = 8 then Q = 2, 3, 7 or 8; if Q = 2 then B + P = 9, if Q = 3 then B - P = 7 for B > P and B - P = -3 for B < P and B can never be equal to P, if Q = 7 then B + P = 11 for B >1 and B + P =1 for B < 1, if Q = 8 then B = P.


It is clear from (i) and (ii) above that if first and last digit are same they can either be 2 or 8 and the second digit B and P will also have to be same. Also the difference of first and last digit i.e. A-Q of any Rare number can only be 0, 1, 4, 5 or 6. The value of Q can never be 1, 4, 6, 9

(iii) The digital root of a Rare number can take the values 2, 5, 8 or 9 only.

The above properties can be summarized in a tabular form as given below:

Properties of Rare numbers
A	Q	B & P
2	2	B = P
4	0	|B - P| = zero or Even
6	0 or 5	|B - P| = Odd
8	2	B + P = 9
8	3	B - P=7 or P - B = 3
8	7	B + P = 11 or B + P = 1
8	8	B = P

The above properties can easily be proved [1] using properties of perfect squares given below.

(i) It is well known that a perfect square can never end in 2, 3, 7 or 8

(ii) Similarly for a number ab...yz to be perfect square, if z = 0 then y = 0, if z =5 then y = 2, if z=1, 4 or 9 then y has to be EVEN and if z = 6 then y is ODD.

(iii) The digital root of a perfect square can only be 1, 4, 7 or 9. Digital root is obtained by adding the sum of the digits of a number until single digit is obtained.

   Computation of Rare Numbers

All non-palindromic Rare numbers up to 10²³ have been computed and it is found that there are 132 such numbers. The list of Rare numbers up to 10²³ is available here.

It is noted that there are only 20 Odd Rare numbers up to 10²³ (Sequence A059755).

I ( Shyam Sunder Gupta from INDIA) have been investigating these numbers since 1989. I have developed computer program in Fortran to compute Rare numbers. In fact with refinement of the code over the years , the program has been made so powerful that all numbers up to 10¹⁴ can be just checked for Rare numbers in less than a minute on Pentium III PC. In few hours I have been able to check up to 10¹⁹ on Pentium core 2 duo E6600.

Since beyond 19 digit, it was faster and easier to use UBASIC for computing Rare Numbers , I developed a program in UBASIC and computed all Rare numbers up to 10²³ . While developing computer programs, properties of Rare numbers as described above have been used.

The main part of algorithm consuming substantial time is for checking for "perfect squares". So what is required is an efficient and fast algorithm to check for integer square roots.

Since inbuilt function of Sqrt takes long time , so to reduce computational time , probabilistic methods using Quadratic Residues can be of great advantage. The choice of moduli is very important. I have used different combinations of moduli in my program to reduce the computational time. It is a very interesting exercise to select combination of moduli for quadratic residue test.

Discovery of new digital properties of the Rare numbers is certainly going to help compute these numbers faster.

Quest for finding higher Rare numbers will certainly bring out improved efficient/fast algorithms for square root checking and also new hidden properties of Rare numbers.

   Conjecture about Rare Numbers

It has been Conjectured that there does not exist any Rare number which is also a Prime i.e. there are no Rare primes. See Shyam's conjecture[5].

Investigation of all Rare numbers up to 10²³ reveals that there is no Rare number which is also prime.

Only odd Rare numbers not ending in 5 can be a candidate for being prime and only 7 such Rare numbers have been found below 10²³ and 6 ending in 7 and one in 3. All these 7 Odd Rare numbers are non-primes.

There appears no reason as to why a Rare number can not be a prime but all efforts to find such a Rare Prime number has not been successful.

It remains to prove or disprove the existence of Rare numbers which are also prime.

   Interesting Observations

• If R consist of even number of digits, then R+R1 must be divisible by 11, So 121 must be a factor of X².
• If R consist of odd number of digits, then R-R1 must be divisible by 11. Since R-R1 is always divisible by 9, So 1089 must be a factor of Y².
• Odd Rare numbers are fewer than even Rare numbers.
• Rare numbers with odd number of digits are fewer than Rare numbers with even number of digits.
• Odd Rare numbers with odd number of digits are still fewer. In fact only three Rare numbers i.e. 6979302951885, 6531727101458000045 and 8200756128308135597 are known up to 10²³, which are odd and also consist of odd number of digits.
• The only one Rare number i.e. 8888070771864228883913 below 10²³ have been found to end in 3.

References:

[1] Gupta, Shyam Sunder, "Systematic computations of rare numbers" The Mathematics Education, Vol. XXXII, No. 3, Sept. 1998.

[2] Gupta, S. S., "Rare numbers" 2001, page 77, June 1990.

[3] Guy, R.  K., "Conway's RATS and other reversals," American Math. Monthly, page 425, May 1989.

[4] Guy, R.  K., "Unsolved problems come of Age" American Math. Monthly, page 908, Dec. 1989.

[5] Rivera, C. "Problems & Puzzles: Conjecture 23. The Shyam's conjecture about the Rare Numbers" http://www.primepuzzles.net/conjectures/conj_023.htm.

[6] Sloane, N. J. A. Sequences A035519, A059755, in "The On-Line Encyclopedia of Integer Sequences." http://oeis.org/.

[7] Wells, David. The Penguin Dictionary of Curious and Interesting Numbers. Middlesex, England: Penguin Books, 1997.

[8] Gerard Schildberger - Rosetta Code Discussion about Rare Numbers.

This page was created in 2001 and updated on 17th June 2020.

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