World!Of
Numbers  WON plate
210 |

[ October 14, 2021 ]
Playing around with recreational examples of various types
like combinations using squares, factorials and other functions
Vinod Muleva (email)

Ex. (4!)2 + (5!)2 = 1012 + 692 + 32 + 22 + 12
5! * 6! * 7! = 7573 + 1183 + 243 + 103 + 33 + 3 * 23
6! – 5! + 4! – 3! + 2! – 1! = 619 (Prime!)
6! + 5! – 4! + 3! – 2! – 1! = 821 (Prime!)
10! – 9! – 8! – 7! – 6! – 5! – 4! – 3! – 2! – 1! = 17942 – 352 – 52 – 12 Vinod gave also some random examples of palindromes
that are the sum of powers.
123212 + 1212 – 1172 + 112 + 62 + 12 = 151808151
100012 + 113112 + 104 + 103 = 227969722

In an effort to structure these terms I propose the following challenge.

Find for all the squares s1 and the 'next' higher square s2 (s2 > s1) so
that their sum is palindromic.

Let me start with some initial terms of s1 + s2 = Pal
12 + 22 = 5
22 + 202 = 404
32 + 182 = 333
42 + 142 = 212
52 + ? = ?
62 + ? = ?
72 + ? = ?
...

Maybe some squares will have an accompanying square s2 far far greater than s1 !!
I wonder if it stays easy to find these s2 squares.

Hope you like my challenge. If so can you do this also for third powers ? In the mean time enjoy also these equations from Vinod
turning the palindromes of the form 10^n–1 into a sum of powers

 9 = 32 99 = 92 + 32 + 32 999 = 312 + 52 +32 + 22 9999 = 992 + 132 + 52 + 22 99999 = 3162 + 112 + 32 + 32 + 22 999999 = 9472 + 3162 + 572 + 92 + 22 9999999 = 31622 + 412 + 82 + 32 + 12 99999999 = 99992 + 1412 + 82 + 72 + 22 999999999 = 316222 + 2212 + 162 + 32 + 32 9999999999 = 999992 + 4472 + 132 + 42 + 22 99999999999 = 2999992 + 999992 + 8942 + 262 + 92 + 22 999999999999 = 9999992 + 14142 + 232 + 82 + 32

I guess these are the minimum number of terms (squares)
with which to express the palindromes ?
If someone can do it with even less terms (powers besides
squares allowed), please let me know. [ December 12, 2021 ]
The ink was not yet dry or Alexandru Petrescu
improved these Vinod equations — with only three terms.
Well done!

 999 = 36 + 35 + 33 9999 = 213 + 36 + 32 99999 = 542 + 433 + 263 999999 = 913 + 902 + 623 9999999 = 1943 + 1393 + 1142 99999999 = 4633 + 3642 + 284 999999999 = 12422 + 12062 + 9993  A000210 Prime Curios! Prime Puzzle  Wikipedia 210 Le nombre 210 ```

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