Does congruence in some modulo count?

### John Cook

Discussion -Is 2013 prime? Obviously not, but can we make it prime by a trick?

A year ago I wrote a blog post claiming that 2012 is prime. The trick was that 2012 is prime when viewed as a number in base 3, base 5, base 13, etc.

I decided to revisit my base trick this year. Is 2013 prime in any base? No. In base n, 2013 represents 2n^3 + n + 3, and this number is composite for every positive n because it can be factored into (n + 1)(2n^2 - 2n + 3).

A year ago I wrote a blog post claiming that 2012 is prime. The trick was that 2012 is prime when viewed as a number in base 3, base 5, base 13, etc.

I decided to revisit my base trick this year. Is 2013 prime in any base? No. In base n, 2013 represents 2n^3 + n + 3, and this number is composite for every positive n because it can be factored into (n + 1)(2n^2 - 2n + 3).

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6 comments

+John Cook I am stealing this for my fb status ..:-)

Related post: Is there anything interesting about 2013?

The smallest uninteresting number and fuzzy logic: http://www.johndcook.com/blog/2012/12/31/fuzzy-logic/

The smallest uninteresting number and fuzzy logic: http://www.johndcook.com/blog/2012/12/31/fuzzy-logic/

Bill Ricker

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http://fullcomment.nationalpost.com/2012/12/31/john-chew-jonathan-kay-how-is-2013-interesting-let-us-count-the-ways/ has interesting observations on number of prime factors in year numbers.

**no**base is even more interesting than prime in odd bases.http://fullcomment.nationalpost.com/2012/12/31/john-chew-jonathan-kay-how-is-2013-interesting-let-us-count-the-ways/ has interesting observations on number of prime factors in year numbers.

+Bill Ricker The alliterative number observation is interesting.

+John Cook Alliterative is interesting, but as a /linguistic/ number property it's dependent on both Base and Language. French linguistic deconstruction of 97 as 4x20+10+7 /Quatre-vingt dix-sept/ would give totally different (and for a guess even scarcer?) alliterative numbers.

I liked Chew's factoid of 2013 being first of a sequence of three 3-factor years, each with the primes being one each of one digit, the next decade, and larger two digit primes.

MMXIII is the last of a run of roman numerals without a subtractive term. In this new century, this occurs with boring regularity each decade.

I liked Chew's factoid of 2013 being first of a sequence of three 3-factor years, each with the primes being one each of one digit, the next decade, and larger two digit primes.

MMXIII is the last of a run of roman numerals without a subtractive term. In this new century, this occurs with boring regularity each decade.

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