a and b are positive integers that `a^(2) + 2b = b^(2) + 2a +5`. The value of b is…………….

Answer

To solve the equation \( a^2 + 2b = b^2 + 2a + 5 \) for positive integers \( a \) and \( b \), we can follow these steps:

### Step 1: Rearranging the equation
Start with the original equation:
\[ a^2 + 2b = b^2 + 2a + 5 \]
Rearranging gives us:
\[ a^2 - 2a - b^2 + 2b - 5 = 0 \]
This can be rewritten as:
\[ a^2 - 2a - (b^2 - 2b + 5) = 0 \]

### Step 2: Consider the quadratic in terms of \( a \)
This equation is quadratic in \( a \). For \( a \) to be an integer, the discriminant must be a perfect square. The discriminant \( D \) is given by:
\[ D = b^2 - 4 \cdot 1 \cdot (-(b^2 - 2b + 5)) = 4(b^2 - 2b + 5) + b^2 = 5b^2 - 8b + 20 \]

### Step 3: Testing integer values for \( b \)
Since \( a \) and \( b \) are positive integers, we can test small values of \( b \) to find corresponding values of \( a \).

1. **For \( b = 1 \)**:
\[ 5(1)^2 - 8(1) + 20 = 5 - 8 + 20 = 17 \quad (\text{not a perfect square}) \]

2. **For \( b = 2 \)**:
\[ 5(2)^2 - 8(2) + 20 = 20 - 16 + 20 = 24 \quad (\text{not a perfect square}) \]

3. **For \( b = 3 \)**:
\[ 5(3)^2 - 8(3) + 20 = 45 - 24 + 20 = 41 \quad (\text{not a perfect square}) \]

4. **For \( b = 4 \)**:
\[ 5(4)^2 - 8(4) + 20 = 80 - 32 + 20 = 68 \quad (\text{not a perfect square}) \]

5. **For \( b = 5 \)**:
\[ 5(5)^2 - 8(5) + 20 = 125 - 40 + 20 = 105 \quad (\text{not a perfect square}) \]

6. **For \( b = 6 \)**:
\[ 5(6)^2 - 8(6) + 20 = 180 - 48 + 20 = 152 \quad (\text{not a perfect square}) \]

7. **For \( b = 7 \)**:
\[ 5(7)^2 - 8(7) + 20 = 245 - 56 + 20 = 209 \quad (\text{not a perfect square}) \]

8. **For \( b = 8 \)**:
\[ 5(8)^2 - 8(8) + 20 = 320 - 64 + 20 = 276 \quad (\text{not a perfect square}) \]

9. **For \( b = 9 \)**:
\[ 5(9)^2 - 8(9) + 20 = 405 - 72 + 20 = 353 \quad (\text{not a perfect square}) \]

10. **For \( b = 10 \)**:
\[ 5(10)^2 - 8(10) + 20 = 500 - 80 + 20 = 440 \quad (\text{not a perfect square}) \]

11. **For \( b = 11 \)**:
\[ 5(11)^2 - 8(11) + 20 = 605 - 88 + 20 = 537 \quad (\text{not a perfect square}) \]

12. **For \( b = 12 \)**:
\[ 5(12)^2 - 8(12) + 20 = 720 - 96 + 20 = 644 \quad (\text{not a perfect square}) \]

13. **For \( b = 13 \)**:
\[ 5(13)^2 - 8(13) + 20 = 845 - 104 + 20 = 761 \quad (\text{not a perfect square}) \]

14. **For \( b = 14 \)**:
\[ 5(14)^2 - 8(14) + 20 = 980 - 112 + 20 = 888 \quad (\text{not a perfect square}) \]

15. **For \( b = 15 \)**:
\[ 5(15)^2 - 8(15) + 20 = 1125 - 120 + 20 = 1025 \quad (\text{not a perfect square}) \]

16. **For \( b = 16 \)**:
\[ 5(16)^2 - 8(16) + 20 = 1280 - 128 + 20 = 1172 \quad (\text{not a perfect square}) \]

17. **For \( b = 17 \)**:
\[ 5(17)^2 - 8(17) + 20 = 1445 - 136 + 20 = 1329 \quad (\text{not a perfect square}) \]

18. **For \( b = 18 \)**:
\[ 5(18)^2 - 8(18) + 20 = 1620 - 144 + 20 = 1496 \quad (\text{not a perfect square}) \]

19. **For \( b = 19 \)**:
\[ 5(19)^2 - 8(19) + 20 = 1805 - 152 + 20 = 1673 \quad (\text{not a perfect square}) \]

20. **For \( b = 20 \)**:
\[ 5(20)^2 - 8(20) + 20 = 2000 - 160 + 20 = 1860 \quad (\text{not a perfect square}) \]

21. **For \( b = 21 \)**:
\[ 5(21)^2 - 8(21) + 20 = 2205 - 168 + 20 = 2057 \quad (\text{not a perfect square}) \]

22. **For \( b = 22 \)**:
\[ 5(22)^2 - 8(22) + 20 = 2420 - 176 + 20 = 2264 \quad (\text{not a perfect square}) \]

23. **For \( b = 23 \)**:
\[ 5(23)^2 - 8(23) + 20 = 2645 - 184 + 20 = 2481 \quad (\text{not a perfect square}) \]

24. **For \( b = 24 \)**:
\[ 5(24)^2 - 8(24) + 20 = 2880 - 192 + 20 = 2708 \quad (\text{not a perfect square}) \]

25. **For \( b = 25 \)**:
\[ 5(25)^2 - 8(25) + 20 = 3125 - 200 + 20 = 2945 \quad (\text{not a perfect square}) \]

### Step 4: Finding a valid pair
Continuing this process, we find that when \( b = 3 \) and \( a = 4 \), we get:
\[ a^2 - 2a - (b^2 - 2b + 5) = 0 \implies 4^2 - 2(4) - (3^2 - 2(3) + 5) = 0 \]
This gives us:
\[ 16 - 8 - (9 - 6 + 5) = 0 \implies 16 - 8 - 8 = 0 \]

Thus, the solution is:
\[ \text{The value of } b \text{ is } 3. \]