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Question 269079: What is the only prime number that is the sum and difference of two primes?
Found 3 solutions by mananth, ikleyn, n2:
Answer by mananth(16949)   (Show Source):
You can put this solution on YOUR website! What is the only prime number that is the sum and difference of two primes?
let one digit be x and the other be y
xy=7 --------------1
x+y=8--------------2
multiply equation 2 by y
xy+y^2 = 8y ---------- 3
subtract 2 from 1
-y^2 = 7-8y
y^2-8y+7=0
y^2-7y-y+7=0
y(y-7)-1(y-7)=0
(y-1)(y-7)=0
y= 7 or 1
y cannot be in the tens place because the number is less than 50.
So the number will be 17
Answer by ikleyn(53751)  (Show Source):
You can put this solution on YOUR website! .
What is the only prime number that is the sum and difference of two primes?
~~~~~~~~~~~~~~~~~~~~~~~
What is written in the post by @mananth, has no any relation/relations to the posed question.
So, my advise to a reader is to ignore the post by @mananth as irrelevant.
Instead, see my correct solution below.
It the prime number 'p' is the sum of two primes p1 and p2
p = p1 + p2, (1)
then it is clear from the parity considerations that one of these numbers, p1 and p2,
must be '2'.
Also, it is clear that if the prime number 'p' is the difference of two primes p1 and p2
p = p3 - p4, (2)
then one of these numbers, namely p4, must be '2'.
Thus we should have
p = p1 + 2, (3)
p = p3 - 2. (4)
From these equations, we see that the difference p3-p1 is 4 (taking the difference of these equations).
One possible example of such triple is 3, 5 and 7.
Indeed, 2 + 3 = 5: the sum of prime numbers 2 and 3 is a prime number 5, and
7 - 2 = 5: the difference of prime numbers 7 and 2 is a prime number 5.
To prove the uniqueness, notice that from equations (3) and (4), the triple must be
p1, p2 =  and p3
with the difference between the adjacent terms of 2, so, p1, p2 and p3 should be three consecutive odd numbers.
Hence, one of p1, p2 or p3 must be multiple of '3', and it makes necessary p1 = 3 (the minimal of the terms is 3).
It means that the triple (3, 5, 7) is the unique triple of this kind.
Thus, prime numbers (2, 3, 5) are the UNIQWUE triple satisfying the problem's conditions.
Solved.
Answer by n2(79)  (Show Source):
You can put this solution on YOUR website! .
What is the only prime number that is the sum and difference of two primes?
~~~~~~~~~~~~~~~~~~~~~~~
It the prime number 'p' is the sum of two primes p1 and p2
p = p1 + p2, (1)
then it is clear from the parity considerations that one of these numbers, p1 and p2,
must be '2'.
Also, it is clear that if the prime number 'p' is the difference of two primes p1 and p2
p = p3 - p4, (2)
then one of these numbers, namely p4, must be '2'.
Thus we should have
p = p1 + 2, (3)
p = p3 - 2. (4)
From these equations, we see that the difference p3-p1 is 4 (taking the difference of these equations).
One possible example of such triple is 3, 5 and 7.
Indeed, 2 + 3 = 5: the sum of prime numbers 2 and 3 is a prime number 5, and
7 - 2 = 5: the difference of prime numbers 7 and 2 is a prime number 5.
To prove the uniqueness, notice that from equations (3) and (4), the triple must be
p1, p2 = and p3
with the difference between the adjacent terms of 2, so, p1, p2 and p3 should be three consecutive odd numbers.
Hence, one of p1, p2 or p3 must be multiple of '3', and it makes necessary p1 = 3 (the minimal of the terms is 3).
It means that the triple (3, 5, 7) is the unique triple of this kind.
Thus, prime numbers (2, 3, 5) are the UNIQWUE triple satisfying the problem's conditions.
Prime number 5 is the ONLY prime number, which is the sum and the difference
of two primes, at the same time. ANSWER
Solved.
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