Project PEN » PEN Solutions Archive http://projectpen.wordpress.com Problems in Elementary Number Theory Sun, 28 Jun 2009 16:03:40 +0000 http://wordpress.com/ en hourly 1 http://www.gravatar.com/blavatar/d36b06741f18f17f831c8e2ac71c290a?s=96&d=http://s.wordpress.com/i/buttonw-com.png Project PEN » PEN Solutions Archive http://projectpen.wordpress.com 20S. Sequences of Consecutive Integers http://projectpen.wordpress.com/2009/06/16/20s-sequences-of-consecutive-integers/ http://projectpen.wordpress.com/2009/06/16/20s-sequences-of-consecutive-integers/#comments Tue, 16 Jun 2009 16:08:13 +0000 danielkohen http://projectpen.wordpress.com/?p=407 ]]>

A37 If n is a natural number, prove that the number

(n + 1)(n + 2)\cdots(n + 10)

is not a perfect square.

A9 Prove that among any ten consecutive positive integers at least one is relatively prime to the product of the others.

O51 Prove the among 16 consecutive integers it is always possible to find one which is relatively prime to all the rest.

Here is the official solution file: PEN A9 A37 O51

]]> http://projectpen.wordpress.com/2009/06/16/20s-sequences-of-consecutive-integers/feed/ 0 danielkohen 19S. Binomial Sum Divisible by Primes http://projectpen.wordpress.com/2009/04/25/19s-binomial-sum-divisible-by-primes/ http://projectpen.wordpress.com/2009/04/25/19s-binomial-sum-divisible-by-primes/#comments Sat, 25 Apr 2009 22:03:21 +0000 cpohoata http://projectpen.wordpress.com/?p=388 ]]>

E16 Prove that for any prime p in the interval \left]n, \frac {4n}{3}\right], p divides

\sum^{n}_{j = 0}{{n}\choose{j}}^{4}.

Here is the official solution file (written by Darij Grinberg, Germany): pen19.pdf.

]]> http://projectpen.wordpress.com/2009/04/25/19s-binomial-sum-divisible-by-primes/feed/ 0 cpohoata 18S.Fractions Mod p and Wolstenholme’s Theorem http://projectpen.wordpress.com/2009/04/08/18sfractions-mod-p-and-wolstenholmes-theorem/ http://projectpen.wordpress.com/2009/04/08/18sfractions-mod-p-and-wolstenholmes-theorem/#comments Wed, 08 Apr 2009 01:30:18 +0000 danielkohen http://projectpen.wordpress.com/?p=360 ]]>

A23 Prove that if

1+\frac{1}{2}+\frac{1}{3}+\cdots+\frac{1}{p-1}

is expressed as a fraction, where p>3 is a prime, then p^2 divides the numerator.

A24 Let p>3 be a prime number and k=\lfloor\frac{2p}{3}\rfloor. Prove that

{p \choose 1}+{p \choose 2}+\cdots+{p \choose k}

is divisible by p^2.

Here is the official solution file: pen18.pdf

]]> http://projectpen.wordpress.com/2009/04/08/18sfractions-mod-p-and-wolstenholmes-theorem/feed/ 0 danielkohen 17S. A Hidden Divisibility http://projectpen.wordpress.com/2009/03/16/334/ http://projectpen.wordpress.com/2009/03/16/334/#comments Mon, 16 Mar 2009 12:15:04 +0000 cpohoata http://projectpen.wordpress.com/2009/03/16/334/ ]]>

A13 Show that for all prime numbers p,

Q(p) = \prod^{p - 1}_{k = 1}k^{2k - p - 1}

is an integer.

Here is the official solution file: pen17.pdf.

]]> http://projectpen.wordpress.com/2009/03/16/334/feed/ 0 cpohoata 16S. Using Quadratic Residues http://projectpen.wordpress.com/2008/12/30/16s-using-quadratic-residues/ http://projectpen.wordpress.com/2008/12/30/16s-using-quadratic-residues/#comments Tue, 30 Dec 2008 05:29:29 +0000 siuhochung http://projectpen.wordpress.com/?p=195 ]]>

C2 The positive integers a and b are such that the numbers 15a+16b and 16a-15b are both squares of positive integers. What is the least possible value that can be taken on by the smaller of these two squares?

Here is the official solution file: pen16.pdf

You can discuss the problem in MathLinks here.

]]> http://projectpen.wordpress.com/2008/12/30/16s-using-quadratic-residues/feed/ 0 siuhochung 15. Exponential Congruence Sequence http://projectpen.wordpress.com/2008/12/21/15-15s/ http://projectpen.wordpress.com/2008/12/21/15-15s/#comments Sun, 21 Dec 2008 16:11:18 +0000 compactorange http://projectpen.wordpress.com/?p=181 ]]>

D5. Prove that for , \underbrace{2^{2^{\cdots^{2}}}}_{n\text{ terms}}\equiv \underbrace{2^{2^{\cdots^{2}}}}_{n-1\text{ terms}}\; \pmod{n}.
D6. Show that, for any fixed integer the sequence 2, \; 2^{2}, \; 2^{2^{2}}, \; 2^{2^{2^{2}}}, \cdots \pmod{n} is eventually constant.

Sorry all, for the delay of the problem 15.  Here goes the solution: pen-15.pdf

]]> http://projectpen.wordpress.com/2008/12/21/15-15s/feed/ 0 compactorange n\geq 2 Click on the formula to view the LaTeX code \,n \geq 1,\, Click on the formula to view the LaTeX code 14S. Different Approaches to an Intuitive Problem http://projectpen.wordpress.com/2008/12/08/14s-different-approaches-to-an-intuitive-problem/ http://projectpen.wordpress.com/2008/12/08/14s-different-approaches-to-an-intuitive-problem/#comments Mon, 08 Dec 2008 15:51:02 +0000 gorofir http://projectpen.wordpress.com/?p=163 ]]>

The fourth problem of the second season of PEN is as follows:

N17. Suppose that a and b are distinct real numbers such that:
a - b, a^{2}-b^{2}, \cdots, a^k-b^k, \cdots are all integers. Show that a and b are integers.

Here is the official solution file: pen14.pdf

You can discuss the problem in MathLinks here.

]]> http://projectpen.wordpress.com/2008/12/08/14s-different-approaches-to-an-intuitive-problem/feed/ 1 gorofir 13S Minimum prime divisors http://projectpen.wordpress.com/2008/11/28/13s-minimum-prime-divisors/ http://projectpen.wordpress.com/2008/11/28/13s-minimum-prime-divisors/#comments Fri, 28 Nov 2008 10:45:29 +0000 compactorange http://projectpen.wordpress.com/?p=154 ]]>

13. [PEN A14 A71]

A14. Let n>1 be an integer. Show that n does not divide 2^n-1.

A71. Determine all integers n>1 such that \frac{2^n +1}{n^2} is an integer.
Here is the official solution file: pen13.pdf

You can also discuss the problems: A14 & A71

]]> http://projectpen.wordpress.com/2008/11/28/13s-minimum-prime-divisors/feed/ 1 compactorange 12S A Generalization of an Identity http://projectpen.wordpress.com/2008/11/10/12s-a-generalization-of-an-identity/ http://projectpen.wordpress.com/2008/11/10/12s-a-generalization-of-an-identity/#comments Mon, 10 Nov 2008 11:00:24 +0000 compactorange http://projectpen.wordpress.com/?p=143 ]]>

12. PEN I10 Show that for all primes p,

\sum_{k=1}^{p-1}{\left\lfloor\frac{k^{3}}{p}\right\rfloor}=\frac{(p+1)(p-1)(p-2)}{4}.

Here is the official solution file: pen12.pdf
You can also discuss the problems here!

]]> http://projectpen.wordpress.com/2008/11/10/12s-a-generalization-of-an-identity/feed/ 5 compactorange 11S Three ways to attack a functional equation http://projectpen.wordpress.com/2008/10/29/11s-three-ways-to-attack-a-functional-equation/ http://projectpen.wordpress.com/2008/10/29/11s-three-ways-to-attack-a-functional-equation/#comments Wed, 29 Oct 2008 11:26:48 +0000 compactorange http://projectpen.wordpress.com/?p=134 ]]>

11. PEN K11 (Canada 2002) Find all functions f: \mathbb{N}_{0}\to \mathbb{N}_{0} such that for all

m, \, n\in \mathbb{N}_{0}:

mf(n)+nf(m)=(m+n)f(m^{2}+n^{2}).

Here, \mathbb{N}_{0} denote the set of all nonnegative integers.

Here is the official solution file: pen11.pdf.
You can also discuss the problems here!

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