1973 AMC 12 Problem 3

Question

The stronger Goldbach conjecture states that any even integer greater than

7

can be written as the sum of two different prime numbers. For such representations of the even number

126,

the largest possible difference between the two primes is

Note: The regular Goldbach conjecture states that any even integer greater than

3

is expressible as a sum of two primes. Neither this conjecture nor the stronger version has been settled.

112

100

92

88

80

For a fixed sum, maximize the difference by minimizing the smaller prime

Test the primes

3,

5,

7,

11,

13,

\ldots

until the complement to

126

is prime

For primes

p\lt q

with

p+q=126,

the difference

q-p=126-2p

is largest when

p

is as small as possible. The complements of

3,

5,

7,

and

11

are

123,

121,

119,

and

115,

none prime. For

p=13,

the complement is

113,

which is prime. Thus the largest difference is

113-13=100.

Therefore, the correct answer is B.

Problem 3 in other years