may I have a solution 4 those:

.Determine those primes which can be written both as a sum and a difference of two primes.

.If a, b are digits; if a >= 1 and if (10a + b) is prime, then (2a – b) is also a prime.

.Prove that there is no infinite arithmatic progression having all its terms primes.

.Determine all primes P for which P+4, P+24, P^2+10, P^2+44 are also primes.

5.prove that n is prime if and only if Φ(n) = n-1, where Φ(n) is Eular's totint function.

6.If p prime, if a Î{1,2,3,….}, then p|(p-1)! a^p + a.

7.If p > 8 is a prime, find (p-5)! = ?(mod p)

8.For every n >1, show that (2^4n + 1)/ 5 is integer.

9.If b|a(a-1) , show that gcd (2a-1,b)=1.

Note: b|a and if (a,c)=1, then (b,c)=1.

10.Find three integers in arithmetic progression such that the product of each two of them is square.

11.Let {Ai,….} i=1,2,…,12 be integers, show that there are two of them whose difference is a multiple of 10.

12. if I have a square with length =3m, 10 point are in the square. Show that there are 2 points with distance < 1.5 m .

13.Concider an infinite chess table, each square has a positive integer in it

a= (b+c+d+e)/4

show that all integers are the same

(use the property of N(natural numbers): every nonempty set of N has a small element).