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).
Please show us what you have done yet. These are typical homework questions that can be solved with a bit of effort. For example, the first one :
What is the Euler function defined as ? It is the number of integers less than n that are relatively prime with n. In the case of n prime, n cannot be divided by any integer less than n, except one (definition of a prime number). This leads to the conclusion : if n is prime, then Φ(n) = n-1.5.prove that n is prime if and only if Φ(n) = n-1, where Φ(n) is Eular's totint function.
Do you see the reasoning ?
Now please go back to your lessons, read them thoroughly and carefully, then try to do the problems (they are really easy, you should be able to do them) and come back with a question if you are stuck on one particular point on a problem (how can I express this as ...)
Please don't post more than two questions in a thread. Otherwise the thread can get convoluted and difficult to follow. Start new threads as necessary for remaining questions. eg. If you have five questions, post two of them in two threads and start a new thread for the remaining one etc.
And if the question has more than two parts to it, it is best to post only that question and its parts in the thread and start a new thread for other questions.
Thread closed.