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### Solution:
First, consider that \(p \times p + 1\) cannot be 2. If \(p \times p + 1 = 2\), then \(p = 1\), but 1 is not a prime number.

Assume \(p\) is an odd number. Then \(p \times p\) is an odd number, and \(p \times p + 1\) is an even number. Also, since \(p \times p + 1\) is a prime number and not 2, this is impossible.

Therefore, \(p\) must be an even number. And since \(p\) is a prime number, \(p = 2\).

Then \(2006 \times p = 2006 \times 2 = 4012\)

\(\boxed{4012}\)

Very high

Everyone is too strong, I admire

First question to find narcissistic numbers simple method:
@echo off && setlocal enabledelayedexpansion
for /l %%i in (100 1 999) do (
set tin=%%i
set /a ton=!tin:~0,1!*!tin:~0,1!*!tin:~0,1!+!tin:~1,1!*!tin:~1,1!*!tin:~1,1!+!tin:~-1!*!tin:~-1!*!tin:~-1!
if !tin! equ !ton! (
echo %%i
)
)
pause>nul

[ Last edited by 3dnowex on 2007-8-20 at 06:08 PM ]

#
Find perfect numbers within 1000.
The so-called perfect number:
For example:
6
2*3=6
2+3+1=6
As long as it meets the above two requirements. (The second condition that +1 is indispensable)

{
??????
}

```
@echo off & setlocal enabledelayedexpansion
for /l %%i in (6 1 1000) do (
title Checking: %%i
set/a sum=1
set/a partnum=%%i/2
for /l %%j in (2 1 !partnum!) do (
set/a var=%%i %% %%j
if !var! EQU 0 set/a sum+=%%j)
if !sum! EQU %%i echo %%i is a perfect number)
pause
```

Re: 126F, come up with a more efficient piece of code:

@echo off & setlocal enabledelayedexpansion

for /l %%i in (6 1 1000) do (

    title Detecting: %%i

    set/a sum=1

    set/a partnum=%%i/2

    if !partnum! gtr 40 set/a partnum=40

    for /l %%j in (2 1 !partnum!) do (

         set/a var=%%i %% %%j

         if !var! EQU 0 (

              set/a tmp=%%i/%%j

              if !tmp! gtr %%j set/a sum+=!tmp!+%%j))

    if !sum! equ %%i echo %%i is a perfect number

)

pause>nul

There is a strange phenomenon:
Using the second piece of code to detect perfect numbers within 1000, there are only three: 6, 28, 496.
After modification, when detecting perfect numbers within 10000, only one more is added: 8128.
Is it because the number is large, so there are more factors, and the sum of factors is generally larger than itself, so the probability of perfect numbers appearing is particularly small?

A bit interesting

Question 25

@echo off

setlocal enabledelayedexpansion

set /a gj=79

for /l %%i in (1,1,79) do (

  set /a j1=200-%%i*4

  set /a j=j1/2

  set /a jg=!j!+%%i

if !jg! equ !gj! echo 鸡=!j! 狗=%%i

)

Learned from the algorithm of the camel hump in youxi01..

Hehe, interesting, learned

Hehe, interesting, learned something

Watch the master learn

28
2*8=16
2+8+1=11

28 is it...