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PEN Global

Anyone who wants to contribute the PEN Global, the translation work of Problems in Elementary Number Theory 2 [ 2009 ] No. 1, please submit your translation to Yimin Ge at \text{\small yimin.ge@gmx.at} together with your LaTeX file, its compiled PDF file and fonts file.

You have to use the TeX file available here. Download gPENv02n1eng.tex. The deadline for your submission is 15th, July..

Before you contact Yimin Ge, please have a look at the PEN Global page at http://projectpen.wordpress.com/global .

Notice: We’ve already find translators: in Vietnamese,  Greek, Spanish, Bangla, Bosnian, Chinese, Croatian, Serbian [Latin] and Serbian [Cyrillic]

As you know, Problems in Elementary Number Theory 2 [ 2009 ] No. 1 is NOW online. The Editors-in-Chief of the next issue are

Daniel Kohen (Argentina)
Cosmin Pohoata (Romania)
Harun Šiljak (Bosnia and Herzegovina)
Peter Vandendriessche (Belgium)

Anyone who wants to be one of the Editors-in-Chief of the fourth and fifth issue of Problems in Elementary Number Theory (2010), APPLY NOW.

Please, send an email to us at pen@problem-solving.be with your brief C.V.

You need to know basic LaTeX skills and of course you have to be fluent with Olympiad-style problems from Elementary number theory.

We are planning to recruit two or three Editors-in-Chief for Problems in Elementary Number Theory 2010. The deadline for application is 31th, July.

from Hojoo Lee (the founder of project PEN)

Is now available here !

This weblication is written by PEN team and contributors:

Alexander Remorov (Canada),
Darij Grinberg (Germany),
Harun Siljak (Bosnia and Herzegovina),
Marin Misur (Croatia).

We also need to thank  two Editors-in-Chief:

Daniel Kohen (Argentina) and Cosmin Pohoata (Romania).

Here goes the table of contents:

1 Problems 1
2 Articles 3
2.1 Three Ways to Attack a Functional Equation . . . . . . . . . . . . . . . . . . . . . 3
2.2 A Generalization of an Identity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8
2.3 Minimum prime divisors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
2.4 Di erent Approaches to an Intuitive Problem . . . . . . . . . . . . . . . . . . . . . 15
2.5 Exponential Congruence Sequence . . . . . . . . . . . . . . . . . . . . . . . . . . . 19
2.6 Using Quadratic Residues . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24
2.7 A Hidden Divisibility . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27
2.8 Fractions Mod p and Wolstenholme’s theorem . . . . . . . . . . . . . . . . . . . . . 29
2.9 A binomial sum divisible by primes . . . . . . . . . . . . . . . . . . . . . . . . . . . 33
2.10 Sequences of Consecutive Integers . . . . . .
1 Problems 1
2 Articles 3
2.1 Three Ways to Attack a Functional Equation . . . . . . . . . . . . . . . . . . . . . 3
2.2 A Generalization of an Identity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8
2.3 Minimum prime divisors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
2.4 Di erent Approaches to an Intuitive Problem . . . . . . . . . . . . . . . . . . . . . 15
2.5 Exponential Congruence Sequence . . . . . . . . . . . . . . . . . . . . . . . . . . . 19
2.6 Using Quadratic Residues . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24
2.7 A Hidden Divisibility . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27
2.8 Fractions Mod p and Wolstenholme’s theorem . . . . . . . . . . . . . . . . . . . . . 29
2.9 A binomial sum divisible by primes . . . . . . . . . . . . . . . . . . . . . . . . . . . 33
2.10 Sequences of Consecutive Integers . . . . . .

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

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.

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.

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}.

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