A256704 Palindromes of the form 4n + 1 that are divisible by 5.

5, 505, 525, 545, 565, 585, 5005, 5225, 5445, 5665, 5885, 50005, 50105, 50205, 50305, 50405, 50505, 50605, 50705, 50805, 50905, 52025, 52125, 52225, 52325, 52425, 52525, 52625, 52725, 52825, 52925, 54045, 54145, 54245, 54345, 54445, 54545, 54645, 54745, 54845

Offset

1,1

Comments

The sequence is infinite.

Creation of the Sequence. The sequence was generated based on the following known results for primitive Pythagorean triples/triangles (PPTs).

Let c represent the hypotenuse of a PPT and a, b be its sides. Without loss of generality, let a be the odd side and b be the even side.

(a) c takes the form 4n+1.

(b) One of a,b,c is divisible by 5.

Let c be divisible by 5. Thus, this sequence represents the palindromic hypotenuse of a PPT that is divisible by 5.

With the exception of 5, the last two digits of each term of this sequence are 05, 25, 45, 65 or 85. Thus, since the terms are palindromes, the first two digits of each term of the sequence are 50, 52, 54, 56 or 58.

Proof. If k is an element of the sequence, then its form must be 4n+1 for some integer n. This means that k-1=4n that is, k-1 is divisible by 4. Since an integer is divisible by 4 if and only if the number composed of its last two digits is divisible by 4, it must be that the number composed of the last two digits of k-1 is divisible by 4. With this we get the desired result.

Except for the terms having one or three digits, the terms of the sequence take the following forms:

(a) 50p05

(b) 52p25

(c) 54p45

(d) 56p65

(e) 58p85

where p is a palindromic number with leading zero digit(s) allowed.

Links

Harvey P. Dale, Table of n, a(n) for n = 1..1000

John Rafael M. Antalan and Richard P. Tagle, Numeric Palindromes in Primitive and Non-Primitive Pythagorean Triples, arXiv:1502.06760 [math.NT], 2015.

Mathematica

palQ[n_] := Block[{d = IntegerDigits@ n}, d == Reverse@ d]; Select[Select[4 Range[100000] + 1, palQ], Mod[#, 5] == 0 &] (* Michael De Vlieger, Apr 08 2015 *)
Select[5*Range[12000], PalindromeQ[#]&&Mod[#, 4]==1&] (* Requires Mathematica version 10 or later *) (* Harvey P. Dale, Jul 09 2018 *)

Prog

(PARI) ispal(n) = (d = digits(n)) && (Vecrev(d) == d);
isok(n) = !(n % 5) && !((n-1) % 4) && ispal(n); \\ Michel Marcus, Apr 08 2015

Crossrefs

Sequence in context: A013305 A114066 A014287 * A083151 A128682 A228093

Adjacent sequences: A256701 A256702 A256703 * A256705 A256706 A256707

Keyword

base,nonn

Author

John Rafael M. Antalan, Apr 08 2015

Extensions

More terms from Michel Marcus, Apr 08 2015

Status

approved