# Enigmatic Code

Programming Enigma Puzzles

## Enigma 1633: Same perfect square

From New Scientist #2799, 12th February 2011 [link]

Harry, Tom and I were each looking to find two 2-digit positive integers which each were the product of two primes and whose difference was also the product of two primes, the six primes all being different.

We each found a different valid solution; in each solution the sum of the two integers was a perfect square. For Harry and Tom it was the same perfect square.

What were my two integers?

[enigma1633]

### One response to “Enigma 1633: Same perfect square”

1. jimrandell 13 December 2011 at 9:44 am

This Python program runs in 34ms.

```from enigma import multiply, factor, is_square, Primes
from itertools import combinations

# find primes < 50
primes = Primes(50)

# find 2-digit products
products = []
for s in combinations(primes, 2):
n = multiply(s) # n is the product of two different primes
if not(9 < n < 100): continue # and is two-digit
products.append((n, s))

# select pairs of the 2-digit products
for s in combinations(products, 2):
t = s[0][0] + s[1][0] # sum
if not is_square(t): continue
d = abs(s[1][0] - s[0][0]) # difference
f = factor(d)
if not(len(f) == 2 and f[0] != f[1]): continue
if not(len(set(f).union(s[0][1], s[1][1])) == 6): continue
print(t, s, d, f)
```

Solution: My integers are 74 and 95.

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