What is the lowest and highest 10 digit prime number?

Question

Rephrased:
What is the lowest 10 digit prime number and what is the highest 10 digit prime number?

Answer 1

Lowest is 1000000007 and highest is 9999999967

Use WIMS to start with 1000000000 and find primes counting up. And start with 9999999999 and count down.

You can verify that 1000000001 is divisible by 11: sum of digits at odd positions = 1, sum of digits at even positions = 1, and this difference 1-1 = 0 is divisible by 11. Use WIMS factorizer to show that:

1000000001 = 7 × 11 × 13 × 19 × 52579, and

1000000003 = 23 × 307 × 141623

and, of course, 1000000003 is divisible by 5.

Counting down, and ignoring obvious multiples of 2, 3 and 5:

9999999997 = 13 × 769230769
9999999991 = 19^2 × 277 × 100003
9999999989 = 7 × 4243 × 336689
9999999983 = 3137 × 3187759
9999999979 = 47 × 3613 × 58889
9999999977 = 11 × 43 × 67 × 315547
(obvious because 99999999 and 77 are both divisble by 11)
9999999973 = 73 × 661 × 207241
9999999971 = 13 × 109 × 1657 × 4259

Answer 2

My most noteworthy rpandigital is 7691523840 = 2^8 * 3^2 * 5 * 7 * 11 * 13 * 23 * 29, which has the most number of distinct positive divisors, i.e. 1728. EDIT: Concatenation of 2-digit primes: 8 different groupings where each can be arranged in 5! ways. Total = 960 rpandigitals. 02 41 53 67 89 02 41 59 67 83 02 47 53 61 89 02 47 59 61 83 05 23 41 67 89 05 23 47 61 89 05 29 41 67 83 05 29 47 61 83 ----- ----- ----- ----- Concatenation of 3-digit primes (9-digit numbers only with non-zero leading digits): Total = 4134. Starting with 103 269 457 and ending with 983 647 521. If you want to make the numbers up to 10 digits with a 1-digit prime, then there are 23328 different ways, such as, for example, 251 3 647 809 and 401 823 659 7. There are 972 groupings where each can be arranged in 4! ways = 23328. 13% of these have a 3 as the 1-digit prime, 62% have a 5, 25% have a 7, and surprisingly, none have a 2. Percentages are approximate. ----- ----- ----- ----- Concatenation of 4-digit primes with 2-digit primes: 1981 different groups where each can be arranged in 3! ways. Total = 11886. e.g. 1049 2657 83 and 9857 6203 41. ----- ----- ----- ----- Concatenation of 5-digit primes: Total = 29352. Starting with 10243 56897 and ending with 98641 72503. EDIT 2: I'm from the old school and am still using Microsoft QuickBASIC because I don't have time to learn another language. Consequently, I program my own libraries. I see that farf and I had the same idea, but as my system is not sophisticated or very fast, I was beaten to the smallest prime problem. It has taken days to get only halfway there, but I'll keep on plugging away to see if I can confirm it and to see if my programming is still up to scratch. Here's something else instead. All sets of three concatenated 3-digit primes (non-zero digits) with square sums : (interesting that three sets have the same sum) 149 257 683 : (six arrangements, sum = 1089 = 33^2) 281 593 647 : (six arrangements, sum = 1521 = 39^2) 283 547 691 : (six arrangements, sum = 1521 = 39^2) 293 587 641 : (six arrangements, sum = 1521 = 39^2) EDIT 3: Sum of two, concatenated, 5-digit primes is a cube : 14087 + 32569 = 36^3 24019 + 86573 = 48^3 72503 + 84961 = 54^3 74561 + 82903 = 54^3 74903 + 82561 = 54^3 76541 + 80923 = 54^3 Biquadrates : 21569 + 83407 = 18^4 23567 + 81409 = 18^4 24659 + 80317 = 18^4 Quintics : there are none. EDIT 4: Product of 4 consecutive integers is : 291 * 292 * 293 * 294 = 7319658024 but I used the computer.

Answer 3

The lowest 10-digit prime number is 1,000,000,007. The highest 10-digit prime number is 9,999,999,967.

Edit: pay no attention to the previous poster. His numbers are composite - 1000000001 factors as 7 * 11 * 13 * 19 * 52579 and 9999999997 factors as 13 * 769230769.

Answer 4

1000000001 is the lowest and the highest is
9999999997