A056810 Numbers whose fourth power is a palindrome.

0, 1, 11, 101, 1001, 10001, 100001, 1000001, 10000001, 100000001, 1000000001, 10000000001, 100000000001

Offset: 1

Author

Robert G. Wilson v, Aug 21 2000

Keywords

• nonn
• base
• more

Comments

Suppose a number is of the form a=10...01 (see A000533) then a^2=10..020..01, so a^2 is always a palindrome. a^3=10..030..030..01, so a^3 is always a palindrome. Similarly we also have a^4=10..040..060..040..01, so a^4 is always a palindrome. However, a^5 is in general not a palindrome, for example 101^5=10510100501. - Dmitry Kamenetsky, Apr 17 2009

The sequence contains no term with digit sum 3. - Vladimir Shevelev, May 23 2011. Proof: There are four possibilities for n:

1) 1+10^k+10^m, 00, 3) 2+10^s, s>0, 4) 3*10^t, t>=0.

In the last two cases n^4 is trivially not a palindrome.

For r>=2, in the second case we have n^4 = (1 + 2*10^r)^4 = 1 + 8*10^r + 4*10^(2*r) + 2*10^(2*r + 1) + 2*10^(3*r) + 3*10^(3*r + 1) + 6*10^(4*r) + 10^(4*r + 1)

which cannot be a palindrome.

If r=1, we have 1+8*10+...9*10^4+10^5 which also is not a palindrome.

The proof for the first case is similar. QED - Vladimir Shevelev, Oct 24 2015

Does every term have the structure 100...0001? Referring to the Simmons (1972) paper, we can also ask, if n is a number whose cube is a palindrome in base 4, must the base-4 expansion of n have the form 100...0001? - N. J. A. Sloane, Oct 22 2015

Links

• G. J. Simmons, Palindromic powers, J. Rec. Math., 3 (No. 2, 1970), 93-98. [Annotated scanned copy]
• G. J. Simmons, On palindromic squares of non-palindromic numbers, J. Rec. Math., 5 (No. 1, 1972), 11-19. [Annotated scanned copy]

Crossrefs

Cf. A186080.

Programs

Mathematica

palQ[n_] := Block[{}, Reverse[idn = IntegerDigits@ n] == idn]; k = 0; lst = {}; While[k < 1000000002, If[ palQ[k^4], AppendTo[lst, k]]; k++]; lst (* Robert G. Wilson v, Oct 23 2015 *)

Python

def ispal(n): s = str(n); return s == s[::-1]
def afind(limit):
    for k in range(limit+1):
        if ispal(k**4): print(k, end=", ")
afind(10000001) # Michael S. Branicky, Sep 05 2021

Extensions

a(11) from Robert G. Wilson v, Oct 23 2015

a(12)-a(13) from Michael S. Branicky, Sep 05 2021