Let's start by listing all the possible 4-digit numbers that can be written as the sum of the squares of its two smallest factors, squared, minus 1:
1681 = (1² + 40²)² - 1
2401 = (1² + 48²)² - 1
3481 = (1² + 58²)² - 1
5041 = (1² + 70²)² - 1
7225 = (1² + 84²)² - 1
10201 = (1² + 100²)² - 1
14641 = (1² + 120²)² - 1
21025 = (1² + 144²)² - 1
29929 = (1² + 172²)² - 1
42849 = (1² + 206²)² - 1
61225 = (1² + 244²)² - 1
87361 = (1² + 292²)² - 1
Next, let's check which of these numbers satisfy the condition that their smallest factors multiply up to be 1 less than a square number. For each number, we'll list its smallest factors and calculate their product:
1681 = 41 × 41; 41 × 40 = 1640 = 40² - 1
2401 = 49 × 49; 49 × 48 = 2352 = 48² - 1
3481 = 59 × 59; 59 × 58 = 3406 = 58² - 1
5041 = 71 × 71; 71 × 70 = 4970 = 70² - 1
7225 = 85 × 85; 85 × 84 = 7140 = 84² - 1
10201 = 101 × 101; 101 × 100 = 10100 = 100² - 1
14641 = 121 × 121; 121 × 120 = 14520 = 120² - 1
21025 = 145 × 145; 145 × 144 = 20880 = 144² - 1
29929 = 173 × 173; 173 × 172 = 29716 = 172² - 1
42849 = 207 × 207; 207 × 206 = 42642 = 206² - 1
61225 = 247 × 247; 247 × 244 = 60268 ≠ n² - 1 for any n
87361 = 295 × 295; 295 × 292 = 86040 ≠ n² - 1 for any n
Thus, only the first nine numbers satisfy both conditions, so the answer is one of those numbers. To determine which one, we can simply check each of them:
For 1681, we have 1681 = 1² + 40² and 40 × 41 = 1640 = 40² - 1, so this works.
For 2401, we have 2401 = 1² + 48² and 48 × 49 = 2352 = 48² - 1, so this works.
For 3481, we have 3481 = 1² + 58² and 58 × 59 = 3406 = 58² - 1, so this works.
For 5041, we have 5041 = 1² + 70² and 70 × 71 = 4970 = 70² - 1, so this works.
For 7225, we have 7225 = 1² + 84² and 84 × 85 = 7140 = 84² - 1, so this works.
For 10201, we have 10201 = 1² + 100² and 100 × 101 = 10100 = 100² - 1, so this works.
For 14641, we have 14641 = 1² + 120² and 120 × 121 = 14520 = 120² - 1, so this works.
For 21025, we have 21025 = 1² + 144² and 144 × 145 = 20880 = 144² - 1, so this works.
For 29929, we have 29929 = 1² + 172² and 172 × 173 = 29716 = 172² - 1, so this works.
Therefore, the answer is 29929.
I am thinking of a 4-digit number. It is equal to the sum of the squares of it's 2 smallest prime factors, squared, minus 1. Its smallest prime factors multiply up to also be 1 less than a square number.
what is the number?