Number Theory and Analysis (1)

Shadow · **Posted:** Tue, 13 Oct 2009 19:46:49 UTC

(Taken from the Pi Mu Epsilon Journal's Fall 2009 Problem Set)

The concatenation of integers, p||q

is defined as $p\cdot\left( 10^{\lfloor\log_{10} q\rfloor +1}\right)+q$ , and it does what you think it does.

i.e. 10||23=1023

likewise

and

Find infinitely many triplets, (a,b,c)

such that

1) $(a\,||\,b)\,||\,c$ is NOT a palindrome
2) $(a^2\,||\,b^2)\,||\,c^2$ IS a palindrome.

Denis · **Posted:** Fri, 23 Oct 2009 17:14:59 UTC

Why can't it be p[10^LEN(q)] + q ?

LEN is LENgth function (in Basic anyway).

Is LEN(n) not "universal" ?

Shadow · **Posted:** Fri, 23 Oct 2009 18:31:12 UTC

Denis wrote:

Why can't it be p[10^LEN(q)] + q ?

LEN is LENgth function (in Basic anyway).

Is LEN(n) not "universal" ?

I've never heard of it myself.

Denis · **Posted:** Sat, 24 Oct 2009 01:33:43 UTC

Well, it's actually ALEN(n) for number of digits;
LEN(n) is bit length.

ALEN(n) seems to be pretty well popular; Google it...

Shadow · **Posted:** Sat, 24 Oct 2009 02:19:09 UTC

Denis wrote:

Well, it's actually ALEN(n) for number of digits;
LEN(n) is bit length.

ALEN(n) seems to be pretty well popular; Google it...

No thanks, I never really enjoy reading anything terribly computer science-y at least not coding-wise (algorithms are cool, but that's secretly number theory anyways). Thanks though!

Denis · **Posted:** Sat, 24 Oct 2009 03:17:27 UTC

Shadow wrote:

Find infinitely many triplets, (a,b,c)

such that
1) $(a\,||\,b)\,||\,c$ is NOT a palindrome
2) $(a^2\,||\,b^2)\,||\,c^2$ IS a palindrome.

I fail to see why 1) is requested:
keep a=1 and c=2 (as example) and ALL resulting a||b||c will not be palindromes,
so infinite solutions; or did I miss something?

2) is sort of fun: first 2 lowest are (apart from trivials 000,111,444,999):
4^2||3^2||31^2 : 169961
96^2||1^2||127^2 : 9216116129

outermeasure · **Posted:** Sat, 24 Oct 2009 03:24:57 UTC

Denis wrote:

Shadow wrote:

Find infinitely many triplets, (a,b,c)

such that
1) $(a\,||\,b)\,||\,c$ is NOT a palindrome
2) $(a^2\,||\,b^2)\,||\,c^2$ IS a palindrome.

I fail to see why 1) is requested:
keep a=1 and c=2 (as example) and ALL resulting a||b||c will not be palindromes,
so infinite solutions; or did I miss something?

2) is sort of fun: first 2 lowest are (apart from trivials 000,111,444,999):
4^2||3^2||31^2 : 169961
96^2||1^2||127^2 : 9216116129

You want both (1) and (2) true at the same time...

Nice problem!

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Number Theory and Analysis (1)

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