These problems illustrate an interesting change in doing mathematics. As posed in the early 1980s, the problems and their solutions required insight and the novel combination of diverse mathematical concepts. However, due to current computer technologies, the problems' solutions take on a different nature now in that they can be determined through brute force or exhasution of available cases.
Your Task: Try to work out these problems BOTH ways, with and without computer/calculator technologies. Monitor your approaches towards solutions. Which approach seems more desireable? Satisfying? Successful? Why...?
[#1] Are there any prime values of p < 2200 for which the equation x3-y3 = p has a solution in positive integers x and y?
[#2] Solve the cryptarithm IS x Now = SOON where IS is a prime number. Also, regard the above equation as a multiplication of positive integers.
[#3] Can different odd digits be placed at the vertices of a triangle and different even digits be placed at the midpoints of its sides in such a way that the 3-digit integers formed by the digits on every side are reversible primes? (NOTE: A reversible prime is prime as written and stil, prime when its digits are reverese.)
[#4] Let N and A be two positive integers such that N > 1 and A ≤ 9. What is peculiar about the number (9A+10)(10N-1)? Prove your assertion(s).
[#5] Among the sum of three consecutive primes greater than 7, locate
- the smallest non-square composite sum;
- the smallest multiple of 5;
- the integer composed of consecutivfe digits; and
- a perfect cube.
[#6] Three 3-digit primes are composed of the nine non-zero digits. Their sum consists of four distinct digits in order of magnitude. Find the primes and their sum.
Source: Various math journals from early 1980s
Hint: You are on your own, except possibly with the aide of a computer and/or calculator.
Solution Commentary: No solution commentary provided, except to note that it is possible to solve all of the problems without using a computer or calculator.
As to the merits of the experience, see the quotations for this week.