RARE NUMBERS
The numbers, which gives a perfect square on adding as well as subtracting its reverse are rare and hence termed as Rare Numbers[1].
If R is a positive integer and R1 is the integer obtained from r by writing its decimal digits in reverse order, then R + R1 and R - R1 both are perfect square then R is termed as Rare Number. 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.
Other Example of Rare numbers are 621770, 281089082, 2022652202, 2042832002, 868591084757, 872546974178,..
(See Neil's sequence A035519).The examples given above are non-palindromic. In case palindromic numbers are also considered then it is noticed that there are infinite numbers of palindromic Rare numbers. For example the numbers in the series 242,20402,2004002, .... are all palindromic Rare numbers.
Unlike palindromic Rare numbers which are infinite, it still remains to find whether non-palindromic Rare numbers are finite or infinite.
All non-palindromic Rare numbers up to 1019 have been found and there are 75 such numbers. The list of Rare numbers up to 1019 is available
here.It is noted that there are only 14 Odd Rare numbers up to 1018 (See Neil's sequence
A059755)Only two more Odd Rare numbers 6531727101458000045 and 8200756128308135597 up to 1019 have been found.
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 conjectureIt may be an interesting exercise to find an example of a Rare number ending in 3, as none is known till date. For some of the properties of Rare numbers and their proof, refer my paper in [1].
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[1]
Systematic computations of rare numbers, Shyam Sunder Gupta, The Mathematics Education, Vol. XXXII, No. 3, Sept. 1998.