|Oct11-06, 09:00 AM||#1|
Euler: Please verify my result !
I got two tasks which I have some trouble with.
A guy has 1770 dollars to shop food for. One bread costs 31 dollars and a jar of jam costs 21 dollars.
How many loafs of bread and jar's of jam can the guy buy?
I'm suppose to calculate it using Euler Algebra
31x + 21y = 1770
Do I solve this by guessing x,y ?
a) Show that an integer, is congruent 3 mod 4, and which has a prime factor which is congruent modulo with 4.
How do I do that?
b) show that infinit many prime numbers are congruent 3 mod 4.
x=5310 and y=-3540
Is one solution of,
Thus, all solutions are:
We require that,
Solving both of these inequalities we get,
Since t is in integer I have,
Corresponding to 3 solution of x and y which are:
Altenate Solution for (1)
I have: .. 31x + 21y .= .1770 .
Then: . . . .31x - 1770 .= .-21y
By definition: . . . 31x .= .1770 (mod 21)
And I have: . . . 10x .= .6 (mod 21)
Divide by 2: . . . . . 5x .= .3 (mod 21)
Multiply by 17: . 17·5x .= .17·3 (mod 21)
and I have: . . . 85x .= .51 (mod 21)
which equals: . . . . .x .= .9 (mod 21)
Hence: . x .= .9 + 21k . for some integer k.
Substitute into : . 31(9 + 21k) + 21y .= .1770 . → . y .= .71 - 31k
There are three solutions.
If k = 0: .(x,y) = (9,71)
If k = 1: .(x,y) = (30,40)
If k = 2: .(x,y) = (51,9)
Which of my two solutions for (1) is best?
Actually I know I can use Dirichlet's Theorem. (One of my favorite mathemations).
But, my professor says I need to prove this without that.
Assume there are finitely many primes of form 4k+3
P1 P2 P3 ... Pn
Form the number,
Prime factorize this number,
N is odd it is either of form 4k+1 or 4k+3
But, N takes the form of4k+3.
Now, if all Qk's in the factorization have form 4k+1
Then, N would have form 4k+1 (As explained in other post).
Which is not true, thus it must have at least one prime Qk which is of form 4k+3.
Since we have equality and the left is divisible by Qk so does the right. But since P1*P2*...*Pn contains all primes of form 4k+3 it is divisible by Qk. But then Qk divide 1!!!! which is impossible. Thus, there most be infinitely many (OR NONE) primes in form of 4k+3.
I know proof is not well formulated, but is there anybody here who maybe could assist me in making it better?
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