EPORNS

**EPORN**s ( **E**qual **P**roduct **O**f **R**eversible **N**umbers) are defined as the numbers
which can be expressed as the product of two reversible numbers in two different
ways [1].

For example:

4030 = 130 x 031 = 310 x 013

144648 = 861 x 168 = 492 x 294

185472 = 672 x 276 = 384 x 483

9949716 = 2583 x 3852 = 1476 x 6741

16746912 = 2556 x 6552 = 4473 x 3744 etc.

The EPORNs have the following properties.

- The smallest number that can be expressed as product of two reversible numbers in two different ways is 2520.

2520 = 120 x 021 = 210 x 012

- The number 2520 is the least common multiple (LCM) of all single digit natural numbers in decimal system. In other words, LCM of all numbers from 1 to 9 gives the smallest EPORN.
- The digital root i.e. ultimate sum of digits of all EPORNs is always 1,4,7 or 9. For example,

2520 = 2 + 5 + 2 + 0 = 9

4030 = 4 + 0 + 3 + 0 = 7

9949716 = 9 + 9 + 4 + 9 + 7 + 1 + 6 = 36 and 3 + 6 = 9

- Only six numbers i.e. 2520, 4030, 5740, 7360,7650 and 9760 are known among the four digit EPORNs. All other EPORNs consist of five or more digits. For example, some of the five digit EPORNs are 10080,12070,13000,14580,14620,17290,63504 etc.
- All known four digit EPORNs are multiples of 10 and all the known five digit EPORNs except 63504 are also multiples of 10. These can be obtained by multiplying the product of two different reversible numbers by 10, with the condition that any of the reversible number must not be a multiple of 10.

For example , 12 x 21 x 10 = 2520, 13 x 31 x 10 = 4030, 14 x 41 x 10 = 5740 etc gives EPORNs but 10 x 01 x 10 = 100, 20 x 02 x 10 =400 etc are not EPORNs as one of the reversible numbers is 10,20 etc, which is a multiple of 10.

- Among the EPORNs, which are not multiples of 10, the smallest is 63504, which is equal to 441 x 144 = 252 x 252, where 252 is a palindromic number so its reverse is same as original number.

All other numbers in this category consist of six or more digits. For example, 101556, 144648, 185472, 5955264, 6794424, 7683984, 8856036, 9523696, 9949716, 11093796, 11807208, 16746912, 18095112, 22218336, 32679864 etc.

- Some of the EPORNs are perfect squares. For example, 63504, 7683984,782432784 etc.

63504 = 252** ^{2} = **441 x 144

7683984 = 2772** ^{2 }= **1584 x 4851

782432784 = 27972** ^{2 }= **15984 x 48951 etc

It can be seen that an EPORN is a perfect square only when it is the square of a palindromic number.

- The square of numbers in the sequence 2772, 27972, 279972, 2799972, 27999972, ..... are all EPORNs so, the number of EPORNs are infinite.

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[1] EPORNS,S. S. Gupta, Science Today, February 1987, India.

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