EPORNS

EPORNs ( Equal Product Of Reversible Numbers) 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.

2520 = 120 x 021 = 210 x 012

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

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.

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.

63504 = 2522 = 441 x 144

7683984 = 27722 = 1584 x 4851

782432784 = 279722 = 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.

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