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We show that $$\sum_{\substack{p_n \in [x, 2x] \\ p_{n+1} - p_n \ge x^{1/2}}}(p_{n+1} - p_n) \ll x^{0.57+\epsilon}$$ and $$\sum_{\substack{p_n \in [x, 2x]\\ p_{n+1} - p_n \ge x^{0.45}}} (p_{n+1} - p_n) \ll x^{0.63+\epsilon},$$ where$p_n$ is the $n$th prime number. The proof combines Heath-Brown's recent workwith Harman's sieve, improving and extending his results. We give applicationsof the results to prime-representing functions, binary digits of primes andapproximation of reals by multiplicative functions.

On large differences between consecutive primes