Abstract.An ongoing search for first occurrence prime gaps continues.An ongoing search for first occurrence prime gaps is being carried out which extends all previous work done on this subject. To date this search has found all such gaps for primes up to 7.263 x 1013. First occurrence prime gaps had previously been known for primes less than 4.444 x 1012 [2]. Several gaps larger than the previously largest gap of 682 (not a first occurrence) found by Weintraub [4] have been found.Computer programs were written in FORTRAN and CAL (Cray Assembly Language) on a CRAY-2 supercomputer.The computation was conducted as follows. A sufficient number of primes were generated to perform a sieve. Odd numbers beginning with 3 were sieved one block at a time, where each block was chosen to contain 40,000,000 numbers based on system resource availability. The even numbers were eliminated during initialization of each block. One number was stored per 64-bit computer word. After sieving each block, the differences between consecutive primes were calculated and stored. This was accomplished by loading 64 elements of the sieved block at a time into a vector register. A 64-bit vector mask was created containing l's for corresponding nonzero values in the vector register. If the vector mask was zero, the next 64 values of the sieved block were loaded into the vector register. If the mask was nonzero, an instruction to count the number of leading zeros was executed to get the offset from the beginning of the vector register for the next prime. A subtraction of the previous prime was done, thus arriving at the gap. The leftmost 1 of the vector mask was cleared and the method repeated, beginning with checking if the vector mask was zero. This was repeated until the entire block was processed. The last prime in each block was saved in order to calculate the difference between that prime and the first prime in the next block to make sure no gaps were missed. The time to perform the sieve for each block, and to calculate all gaps generated, was about 10.5 seconds for numbers in the range of 7.2 x 1013. The largest prime in the last block processed to date is 72635119999997, so the table of first occurrence prime gaps is complete to that prime.

We have shown by machine proof that F22 = 22 +1 is composite. In addition, we reenacted Young and Buell's 1988 resolution of F20 as composite, finding agreement with their final Selfridge-Hurwitz residues. We also resolved the character of all extant cofactors of Fn , n < 22, finding no new primes, and ruling out prime powers.

Abstract.The twentieth Fermât number, F20 = 22 +1, has been proven composite by machine computation.The Fermât numbers are the numbers Fn = 22" + 1, orginally conjectured by Fermât to be prime for all n. In fact, only for n equal to 0 through 4 are they known to be prime, and small factors of Fg, Fn, F12, Fis, Fie, have been known for some time. As part of a long-term test of the hardware reliability of the Cray-2 supercomputer at the Supercomputing Research Center, the authors proved that F20 = 22 +1, which had been the smallest Fermât number of unknown character, is composite. The test for compositeness was the standard technique of Pépin The result of the computation on the Cray-2 has been verified by performing the same computation on a Cray X-MP belonging to Cray Research. The total computation time on the Cray-2 was about 10 CPU days; the time on the Cray X-MP was 82 hours. Both programs ran as single-processor programs on any available CPU of the respective machines; the ability of either computer to run in parallel on multiple CPU's was not used. The time needed to test Fn, for n in the range 10 through 20, is just slightly more than four times the time needed to test Fn_i: The number of multiplications doubles in incrementing n, and the time required for each multiplication doubles, being dependent almost entirely on the length of the operands. Our programs would thus determine the character of F22, which is now the smallest Fermât number of unknown character, in a little more than 16 times the time needed for our computation on i^o-The table below summarizes what is now known about the Fermât numbers for n less than or equal to 22. A status list for larger n appears in [1].This computation, roughly one million squarings modulo a one million bit number, would be impossible to do even on supercomputers without fast Fourier transform techniques for integer multiplication.Since one reason for performing this computation was to verify hardware reliablity and not to minimize the execution time, the program was written entirely in Cray Fortran and called Cray library functions for the FFT's. The program itself was very simple and only about 200 lines long, much of which was used for checkpointing and restarting the program. The program was called into execution every time the Cray-2 was restarted, and so

For many years, ethernet has been the mainstay for TCP/IP and local area networking, and issues specific to wide area and long haul networks have not been adequately addressed. The advent of FDDI and HIPPI standards, which are, respectively, one and two orders of magnitude faster then Ethernet, and high speed cross country links, are causing what used to be experimental issues to become everyday problems. This paper will cover some of these issues, as they relate to the TCP/IP protocols, and the work that has happened at Cray Research in the development of the UNICOS&dagger; operating system to address these issues.

Abstract.The twentieth Fermât number, F20 = 22 +1, has been proven composite by machine computation.The Fermât numbers are the numbers Fn = 22" + 1, orginally conjectured by Fermât to be prime for all n. In fact, only for n equal to 0 through 4 are they known to be prime, and small factors of Fg, Fn, F12, Fis, Fie, have been known for some time. As part of a long-term test of the hardware reliability of the Cray-2 supercomputer at the Supercomputing Research Center, the authors proved that F20 = 22 +1, which had been the smallest Fermât number of unknown character, is composite. The test for compositeness was the standard technique of Pépin The result of the computation on the Cray-2 has been verified by performing the same computation on a Cray X-MP belonging to Cray Research. The total computation time on the Cray-2 was about 10 CPU days; the time on the Cray X-MP was 82 hours. Both programs ran as single-processor programs on any available CPU of the respective machines; the ability of either computer to run in parallel on multiple CPU's was not used. The time needed to test Fn, for n in the range 10 through 20, is just slightly more than four times the time needed to test Fn_i: The number of multiplications doubles in incrementing n, and the time required for each multiplication doubles, being dependent almost entirely on the length of the operands. Our programs would thus determine the character of F22, which is now the smallest Fermât number of unknown character, in a little more than 16 times the time needed for our computation on i^o-The table below summarizes what is now known about the Fermât numbers for n less than or equal to 22. A status list for larger n appears in [1].This computation, roughly one million squarings modulo a one million bit number, would be impossible to do even on supercomputers without fast Fourier transform techniques for integer multiplication.Since one reason for performing this computation was to verify hardware reliablity and not to minimize the execution time, the program was written entirely in Cray Fortran and called Cray library functions for the FFT's. The program itself was very simple and only about 200 lines long, much of which was used for checkpointing and restarting the program. The program was called into execution every time the Cray-2 was restarted, and so