document.write( "Question 81759This question is from textbook Strategies for Problem Solving Workbook
\n" ); document.write( ": Can you please help me with this problem I couldn't find a strategy to answer the question. \"Find two prime numbers that, if multiplied, would generate a 400-digit number.\" Please help as soon as you can. Thank you \n" ); document.write( "
Algebra.Com's Answer #58604 by Edwin McCravy(20056)\"\" \"About 
You can put this solution on YOUR website!
\r\n" );
document.write( "Can you please help me with this problem I couldn't find a strategy to answer\r\n" );
document.write( "the question. \"Find two prime numbers that, if multiplied, would generate a \r\n" );
document.write( "400-digit number.\" Please help as soon as you can. Thank you\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "You couldn't be expected to answer this from scratch without a super-computer.\r\n" );
document.write( "However, you can go online and see what large primes have been discovered.\r\n" );
document.write( "\r\n" );
document.write( "You can go to this website at the University of Tennessee at Martin\r\n" );
document.write( "\r\n" );
document.write( "http://primes.utm.edu/mersenne/index.html\r\n" );
document.write( "\r\n" );
document.write( "and find a list of what are known as \"Mersenne\" primes. These are primes of the\r\n" );
document.write( "form 2p-1.  We find that the largest one of these that has less than 400 digits\r\n" );
document.write( "is when p = 1279, which has 386-digit,\r\n" );
document.write( "\r\n" );
document.write( "so 21279-1 is a 386-digit prime. \r\n" );
document.write( "\r\n" );
document.write( "Now we know that if we multiply two positive integers together, the number\r\n" );
document.write( "of digits in the product will either be the sum of the numbers of digits in\r\n" );
document.write( "the two integers multiplied or one less than that sum. I'll just demonstrate\r\n" );
document.write( "this fact with a 4 and a 5 digit number: \r\n" );
document.write( "\r\n" );
document.write( "The largest 4 digit number times the largest 5 digit number is\r\n" );
document.write( "\r\n" );
document.write( "9999×99999 = 999890001 which has 9 digits, the sum of 4 and 5.\r\n" );
document.write( "\r\n" );
document.write( "The smallest 4 digit number times the smallest 5 digit number is\r\n" );
document.write( "\r\n" );
document.write( "1000×10000 = 10000000 which is 8 digits, one less than the sum\r\n" );
document.write( "of 4 and 5.\r\n" );
document.write( "\r\n" );
document.write( "And we know the product of any 4-digit number and any 5-digit number\r\n" );
document.write( "will be between these, and thus will either be an 8 or 9-digit number.\r\n" );
document.write( "\r\n" );
document.write( "Similarly the product of any 386-digit number and any 14-digit number\r\n" );
document.write( "will either be a 399 or 400-digit number.\r\n" );
document.write( "\r\n" );
document.write( "Also similarly, the product of any 386-digit number and any 15-digit number\r\n" );
document.write( "will either be a 400 or 401-digit number.\r\n" );
document.write( "\r\n" );
document.write( "Now since we have found a prime with 386 digits, we need either a\r\n" );
document.write( "14 or 15 digit prime, but we don't know which. We can write this\r\n" );
document.write( "inequality:\r\n" );
document.write( "\r\n" );
document.write( "(21279)N < (21279-1)N < (21280)N\r\n" );
document.write( "\r\n" );
document.write( "take the common logs (logs base 10) of \r\n" );
document.write( "all three sides. The three will be in the same order \r\n" );
document.write( "since the log is a strictly increasing function.\r\n" );
document.write( "\r\n" );
document.write( "Furthermore the next higher integer to log(N) is\r\n" );
document.write( "the number of digits in N\r\n" );
document.write( "\r\n" );
document.write( "I'll demonstrate this with the smallest and largest\r\n" );
document.write( "4-digit numbers:\r\n" );
document.write( "\r\n" );
document.write( "log(1000) = 3 and log(9999) = 3.000056568\r\n" );
document.write( "\r\n" );
document.write( "Both 1000 and 9999 are 4-digit numbers and the next\r\n" );
document.write( "higher integer to both their logs is 4.\r\n" );
document.write( "\r\n" );
document.write( "(21279)N < (21279-1)N < (21280)N\r\n" );
document.write( "\r\n" );
document.write( "So take the log of all three sides: \r\n" );
document.write( "\r\n" );
document.write( "1279 log(2) + log(N) < log(PRODUCT) < 1280 log 2 + log(N)\r\n" );
document.write( "\r\n" );
document.write( "385.0173645 + log(N) < log (PRODUCT) < 385.3183944 + log(N)\r\n" );
document.write( "\r\n" );
document.write( "If we can guarantee the left side to be > 399 and the right side to be\r\n" );
document.write( "less than 400 then we can be assured that the PRODUCT will have 400\r\n" );
document.write( "digits\r\n" );
document.write( "\r\n" );
document.write( "For the left side to be > 399\r\n" );
document.write( "\r\n" );
document.write( "385.0173645 + log(N) > 399\r\n" );
document.write( "              log(N) > 13.9826355\r\n" );
document.write( "                  N  > 9.6 × 1013\r\n" );
document.write( "\r\n" );
document.write( "For the left side to be < 400\r\n" );
document.write( "\r\n" );
document.write( "385.0173645 + log(N) < 400\r\n" );
document.write( "              log(N) < 14.9826355\r\n" );
document.write( "                  N  < 9.6 × 1014\r\n" );
document.write( "\r\n" );
document.write( "So any prime between 9.6 × 10 13 and 9.6 × 10^14 will do.\r\n" );
document.write( "\r\n" );
document.write( "On this AT&T research website: \r\n" );
document.write( "\r\n" );
document.write( "http://akpublic.research.att.com/~njas/sequences/table?a=3617&fmt=4\r\n" );
document.write( "\r\n" );
document.write( "there is a list of the smallest n-digit primes and we find the\r\n" );
document.write( "smallest 15 digit prime is 100000000000031\r\n" );
document.write( "\r\n" );
document.write( "That number is between 9.6 × 10 13 and 9.6 × 10^14.\r\n" );
document.write( "\r\n" );
document.write( "So two primes, the product of which has 400 digits are\r\n" );
document.write( "\r\n" );
document.write( "21279 - 1 and 100000000000031    \r\n" );
document.write( "\r\n" );
document.write( "Edwin

\n" ); document.write( " \n" ); document.write( "
\n" );