Palindromic Square 69696 = 264^{2}

From *“Keys to Infinity”* by Clifford A. Pickover, Ed. Wiley 1995, page 159,160.

**69696** is a remarkable palindrome.

Not only is it a palindromic square 264^{2} (see also WONplate 24),

not only has it exactly 10 palindromic prime factors **2**^{6} x **3**^{2} x **11**^{2},

it is also the largest ' smoothly undulating ' (of the form ababab...) square known to humanity.

Visit Sloane's database to find out the other smaller undulating squares at A016073.

I submitted a variant namely ' undulating primes ', watch out for entry A032758.

A less known fact (!?) is that **69696** is also the largest palindromic substring in 8^{117}.

4586997231980143023221641790604173881593129978336562247475

17767877384575217**69696**16140037106220251373109248

Thanks to G. L. Honaker, Jr. we know that **69696** is also the sum of a Twin Prime Pair.

**69696** = 34847 + 34849

[ See Sloane's A037076 and Prime Curio! 69696 ].

Garland reports the following from *“NUMBERS: Fun & Facts”* by J. Newton Friend (1954) :

**69696** is the product of two palindromes. Thus **69696** = **6336** x **11**.

Note **6336** is unusual in that **6336** = **8** x (**63** + **36**) x **8** a nice palindromic expression.

[ *October 24, 2005* ]

Another quality of **69696** is that when it is **squared** and then **doubled**

the outcome will be a **pandigital number** namely **9715064832** !

**2** * **69696** ^ **2**

Note again the palindromic expression itself !

(There exists another 5-digit palindrome with this property. Can you trace it ?)

Note also that this pandigital equals 264^{4} + 264^{4}.

Did you know that

**69696** is the product of **192 * 363**

and that **192 + 363** equals **555** which is a nice repdigit !

( Bonus : **363 – 192** = **171** yet another palindrome popping up ! )

[ *April 30, 2022* ]

Alexandru Petrescu, from time to time obsessed with nine- and/or pandigitals,

challenged himself by searching for palindromes

which when multiplied with **69696** produces

these nine- and pandigital numbers.

He found 3 ninedigitals and 4 pandigitals.

**69696** * 6996 = 487593216

**69696** * 11811 = 823179456

**69696** * 12821 = 893572416

**69696** * 24242 = 1689570432

**69696** * 42424 = 2956783104

**69696** * 62526 = 4357812096

**69696** * 65756 = 4582930176